FUNÇÕES CONVEXAS COM APLICAÇÕES EM PROBLEMAS DE OLIMPÍADAS DE MATEMÁTICA.

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚

❋❯◆➬Õ❊❙ ❈❖◆❱❊❳❆❙ ❈❖▼ ❆P▲■❈❆➬Õ❊❙ ❊▼

P❘❖❇▲❊▼❆❙ ❉❊ ❖▲■▼P❮❆❉❆❙ ❉❊ ▼❆❚❊▼➪❚■❈❆

❱❛❧❡ss❛ ❩❛✐❣❧❛ ❋❛✉st✐♥♦ ❙♦✉s❛ ❈❛r✈❛❧❤♦

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❱❛❧❡ss❛ ❩❛✐❣❧❛ ❋❛✉st✐♥♦ ❙♦✉s❛ ❈❛r✈❛❧❤♦

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✿

❋❯◆➬Õ❊❙ ❈❖◆❱❊❳❆❙ ❈❖▼ ❆P▲■❈❆➬Õ❊❙ ❊▼

P❘❖❇▲❊▼❆❙ ❉❊ ❖▲■▼P❮❆❉❆❙ ❉❊ ▼❆❚❊▼➪❚■❈❆

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ à ❈♦♦r❞❡♥❛çã♦ ❆❝❛❞ê♠✐❝❛ ■♥st✐t✉❝✐♦♥❛❧ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í✱ ♦❢❡r❡❝✐❞♦ ❡♠ ❛ss♦❝✐❛çã♦ ❝♦♠ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ♠❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ❏✉r❛♥❞✐r ❞❡ ❖❧✐✈❡✐r❛ ▲♦♣❡s

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❈❛r✈❛❧❤♦✱ ❱✳❩

①①①① ❋❯◆➬Õ❊❙ ❈❖◆❱❊❳❆❙ ❈❖▼ ❆P▲■❈❆➬Õ❊❙ ❊▼ P❘❖❇▲❊▼❆❙ ❉❊ ❖▲■▼P❮❆❉❆❙ ❉❊ ▼❆❚❊▼➪❚■❈❆✳

❱❛❧❡ss❛ ❩❛✐❣❧❛ ❋❛✉st✐♥♦ ❙♦✉s❛ ❈❛r✈❛❧❤♦ ✕ ❚❡r❡s✐♥❛✿ ✷✵✶✸✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏✉r❛♥❞✐r ❞❡ ❖❧✐✈❡✐r❛ ▲♦♣❡s✳

❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞r❡ ❆r❛ú❥♦ ❞❡ ❙♦✉s❛✳

✶✳ ▼❛t❡♠át✐❝❛

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✐

❉❡❞✐❝❛tór✐❛✿

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ s❡♠♣r❡ ♠❡ ❞❡r❛♠ s✉♣♦rt❡ ❛♦ ❧♦♥❣♦ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❡s❝♦❧❛r ❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳

❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠❡✉ ❡s♣♦s♦ ❱❛❧t❡r❝✐♦ ❆❧♠❡✐❞❛✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❞❡❞✐❝❛çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ✐♥❝❡♥t✐✈♦ ❡ ❝♦♠♣r❡❡♥sã♦✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❝♦❧❡❣❛s ❞♦ P❘❖❋▼❆❚ ❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛ ♠❡✉ ❣r✉♣♦ ❞❡ ❡st✉❞♦✱ ❝♦♥st✐t✉í❞♦ ♣♦r ❊❞✐✈❛♥✱ ❊t❤✐❛♠❛r❛✱ ❍é❧❞❡r✱ ❏❛♥✐❡❧✱ ▼❛r❝♦s ◆❡r②✱ ◆❛s❝✐♠❡♥t♦✱ P❛✉❧♦ ❡ ❱❛❧t❡r❝✐♦✳ ❙❡♠ ❛ ❛❥✉❞❛ ❞❡❧❡s✱ ❝❡rt❛♠❡♥t❡ ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦ ❝❤❡❣❛r ❛té ❛q✉✐✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦ ♣r♦❢❡ss♦r ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞r❡ ✭❈♦♦r✐❡♥t❛❞♦r✮✱ ♣❡❧❛ ✐❞é✐❛ ❞♦ t❡♠❛ ❡ ♣❡❧❛ ❝♦♥❝❡ssã♦ ❞♦ ♠❛t❡r✐❛❧ ❞❡ ❡st✉❞♦ ♣❛r❛ ❛ ❞✐ss❡rt❛çã♦✱ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❏✉r❛♥❞✐r ❞❡ ❖❧✐✈❡✐r❛ ▲♦♣❡s ✭❖r✐❡♥t❛❞♦r✮✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛♣♦✐♦ ❡ ♦r✐❡♥t❛çã♦✱ ❡ ❛ t♦❞♦s ❛q✉❡❧❡s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❡ ❢♦r♠❛ ❡❢❡t✐✈❛ ♥♦ ♠❡str❛❞♦✳

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❉r✳ ❆r♥❛❧❞♦ ❙✐❧✈❛ ❇r✐t♦ ❡ ❉r✳ ❇❛r♥❛❜é P❡ss♦❛ ▲✐♠❛✱ ♣♦r ❛❝❡✐t❛r❡♠ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❞❡ ❞❡❢❡s❛ ❞❛ ❞✐ss❡rt❛çã♦✳

❆❣r❛❞❡ç♦ ❛ ❈❛♣❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❛ ♠✐♠ ❝♦♥❝❡❞✐❞♦✳

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✐✐✐

✏P❛r❛ ❡✈♦❧✉✐r ❡ s♦♠❛r ❝♦♥q✉✐st❛s é ♣r❡❝✐s♦ ❛❞✐❝✐♦♥❛r ♣❡rs✐stê♥❝✐❛ ❡♠ t✉❞♦✑✳

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❘❡s✉♠♦

❖ ❡st✉❞♦ ❝❡♥tr❛❧ ❞❡ss❛ ❞✐ss❡rt❛çã♦ ❛❜♦r❞❛rá ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ❢✉♥çõ❡s ❝♦♥✲ ✈❡①❛s ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳ ❯s❛r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çõ❡s ❝♦♥✈❡①❛s ♣❛r❛ ♠♦str❛r r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s✱ ♠é❞✐❛s ❡ ♥♦r♠❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❉❡s✐❣✉❛❧✲ ❞❛❞❡ ❞❡ ❏❡♥s❡♥✱ ●❡♥❡r❛❧✐③❛çã♦ ❞❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s✱ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▼✐♥❦♦✇s❦✐✱ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ ❡ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✳

❊ss❡ t❡♠❛ é ❜❛st❛♥t❡ ❛♣r♦♣r✐❛❞♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ❛❧✉♥♦s q✉❡ ❜✉s❝❛♠ ❢❡rr❛♠❡♥t❛s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❞❡ ❖❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✳ ❯s❛r❡♠♦s✱ ♣♦✐s✱ ❛❧❣✉♥s r❡✲ s✉❧t❛❞♦s ❛❜♦r❞❛❞♦s ♥❡st❛ ❞✐ss❡rt❛çã♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❡ss❡♥❝✐❛❧ à r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♦❧í♠♣✐❝♦s ❞❡ ✈ár✐♦s ♥í✈❡✐s ❞❡ ❞✐✈❡rs❛s ♦❧✐♠♣í❛❞❛s✳ ❖s r❡s✉❧t❛❞♦s ❝♦♥t✐❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ❢♦r❛♠ ❡①tr❛í❞♦s ❞♦s ❛rt✐❣♦s ❬✷❪✱ ❬✼❪✱ ❬✽❪✱ ❬✾❪ ❡ ❬✶✵❪✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ❉❡s✐❣✉❛❧❞❛❞❡s✱ ❋✉♥çõ❡s ❈♦♥✈❡①❛s✱ ▼é❞✐❛s✱ ◆♦r♠❛s✱ ❖❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✳

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❆❜str❛❝t

❚❤❡ ❝♦r❡ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ st✉❞② ✇✐❧❧ ❛❞❞r❡ss t❤❡ ❜❛s✐❝s ♦❢ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s ✐♥ ❛ r❡❛❧ ✈❛r✐❛❜❧❡✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❝♦♥❝❡♣ts ♦❢ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s t♦ s❤♦✇ ✐♠♣♦rt❛♥t r❡s✉❧ts ♦❢ ✐♥❡q✉❛❧✐t✐❡s✱ ❛✈❡r❛❣❡s ❛♥❞ st❛♥❞❛r❞s✱ s✉❝❤ ❛s✿ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ♦❢ ❏❡♥s❡♥✱ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ♦❢ ❛✈❡r❛❣❡s✱ ❢r♦♠ ▼✐♥❦♦✇s❦✐✱ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ❛♥❞ ❍ö❧❞❡r✬s ✐♥❡q✉❛❧✐t②✳ ❚❤❛t✬s ❛♥ ✐♥t❡r❡st✐♥❣ t❤❡♠❡✱ ♣❛rt✐❝✉❧❛r❧② t♦ st✉❞❡♥ts ✇❤♦ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r t♦♦❧s t♦ r❡s♦❧✈❡ ✐ss✉❡s ♦❢ ▼❛t❤❡♠❛t✐❝s ❖❧②♠♣✐❝s✳ ❲❡ ✉s❡ t❤❡r❡❢♦r❡ s♦♠❡ r❡s✉❧ts ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ❛s ❛♥ ❡ss❡♥t✐❛❧ t♦♦❧ ✐♥ tr♦✉❜❧❡s❤♦♦t✐♥❣ ❖❧②♠♣✐❝ ✈❛r✐♦✉s ❧❡✈❡❧s ❛♥❞ ✈❛r✐♦✉s ❖❧②♠♣✐❛❞s✳ ❚❤❡ ♠❛✐♥ r❡s✉❧ts ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s ♣❛♣❡r ✇❡r❡ ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛rt✐❝❧❡s✿ ❬✷❪✱❬✼❪ ❬✽❪✱ ❬✾❪ ❛♥❞ ❬✶✵❪✳

❑❡②✇♦r❞s✿ ■♥❡q✉❛❧✐t✐❡s✱ ❈♦♥✈❡① ❋✉♥❝t✐♦♥s✱ ▼❡❛♥s✱ ❙t❛♥❞❛r❞s✱ ▼❛t❤ ❖❧②♠♣✐❝s✳

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❙✉♠ár✐♦

❘❡s✉♠♦ ✐✈

❆❜str❛❝t ✈

✶ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✹

✶✳✶ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ❉❡r✐✈á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷ ▼é❞✐❛s ✷✶

✷✳✶ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛ ✭▼❍✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ✭▼●✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ✭▼❆✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ▼é❞✐❛ ◗✉❛❞rát✐❝❛ ✭▼◗✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻ ♣✲▼é❞✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✼ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸ ◆♦r♠❛s ✷✾

✸✳✶ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▼✐♥❦♦✇s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✹ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹ Pr♦❜❧❡♠❛s ❞❡ ❖❧✐♠♣í❛❞❛s ✸✻

✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✽

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❙✉♠ár✐♦ ✈✐✐

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■♥tr♦❞✉çã♦

❆ ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❝♦♠♣❡t✐çã♦ ✐♥s♣✐r❛❞❛ ♥♦s ❥♦❣♦s ♦❧í♠♣✐❝♦s✱ q✉❡ ♣♦r s✉❛ ✈❡③ sã♦ ✐♥s♣✐r❛❞♦s ♥♦s ❢❡st✐✈❛✐s ❡s♣♦rt✐✈♦s q✉❡ ♦s ❣r❡❣♦s r❡❛❧✐③❛✈❛♠ ♥❛ ❛♥t✐❣❛ ➱❧✐❞❛✱ ❡♠ ❤♦♥r❛ ❛♦ ❞❡✉s ❩❡✉s ❡ ❞❡ ♦✉tr♦s ❞❡✉s❡s q✉❡ ❤❛❜✐t❛✈❛♠ ♦ ❖❧✐♠♣♦✳ ❱❡r ❬✶✸❪✱ ❬✶✹❪ ❡ ❬✶✺❪✳

Pr✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞❛s ♦❧✐♠♣í❛❞❛s✿

• ■♥❞✉③✐r ♥♦s ❥♦✈❡♥s ♦ ❣♦st♦ ❡ ♦ ♣r❛③❡r ❞❡ ❡st✉❞❛r ▼❛t❡♠át✐❝❛✳

• ❊st✐♠✉❧❛r ♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ♥♦

❊♥s✐♥♦ ▼é❞✐♦✳

• ❉✐s♣♦♥✐❜✐❧✐③❛r ❛♦s ❡st✉❞❛♥t❡s ❡ ♣r♦❢❡ss♦r❡s ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡st✐♠✉❧❛♥t❡s

❡ ❞❡s❛✜❛❞♦r❡s✳

❆ ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❞✐s♣✉t❛ ❡♥tr❡ ♦s ❥♦✈❡♥s✱ ❞❡ ❝❛rát❡r ✐♥t❡❧❡❝t✉❛❧✱ ✉♠ t♦r♥❡✐♦ ♦♥❞❡ ❛s ❛r♠❛s ❞♦s ♣❛rt✐❝✐♣❛♥t❡s sã♦✿ ❛ ✐♥t❡❧✐❣ê♥❝✐❛✱ ❛ ❝r✐❛t✐✈✐❞❛❞❡✱ ❛ ✐♠❛❣✐♥❛çã♦ ❡ ❛ ❞✐s❝✐♣❧✐♥❛ ♠❡♥t❛❧✳

◆❛ ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ♦s ❡st✉❞❛♥t❡s sã♦ ❞✐✈✐❞✐❞♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❛♥♦ q✉❡ ❝✉rs❛ ♥❛ s✉❛ ❡s❝♦❧❛ ❡ ❝♦♥❝♦rr❡♠ ❡①♣❡r✐♠❡♥t❛♥❞♦ ♦ ♣r❛③❡r ❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ✐♥tr✐✲ ❣❛♥t❡s✳ ❊st❡ t✐♣♦ ❞❡ ❛t✐✈✐❞❛❞❡ ✐♥t❡❧❡❝t✉❛❧✱ q✉❡ ✈❛❧♦r✐③❛ ❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ♦ s❛❜❡r✱ é ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❝✐✈✐❧✐❞❛❞❡ ❡ ❛✈❛♥ç♦ ❝✉❧t✉r❛❧✳ ❆ ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡ ❝♦♠♣r♦✈❛ q✉❡ ❛s s♦❝✐❡❞❛❞❡s ♠❛✐s ❞❡s❡♥✈♦❧✈✐❞❛s tê♠ ❝✉❧t✐✈❛❞♦ ❡ss❡ s❡♥t✐♠❡♥t♦ ❞❡ r❡s♣❡✐t♦ ♣❡❧❛s ✈✐tór✐❛s ❞♦ ❡s♣ír✐t♦✳

❆ r❡❛❧✐③❛çã♦ ❞❡ ❖❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛ ♥♦ ♠✉♥❞♦ é ✉♠ ❛❝♦♥t❡❝✐♠❡♥t♦ q✉❡ ❞❛t❛ ❞♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡✱ s❡♥❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛ ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛ ♦❝♦rr❡✉ ♥♦ ▲❡st❡ ❊✉r♦♣❡✉✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♥❛ ❍✉♥❣r✐❛✱ ♥♦ ❛♥♦ ❞❡ ✶✽✾✹✱ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ ▼✐♥✐str♦ ❞❛ ❊❞✉❝❛çã♦ ❞❛ ❍✉♥❣r✐❛✱ ❏ós❡❢ ❑ürs❝❤á❦✱ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ♠❡♠❜r♦ ❞❛ ❆❝❛❞❡♠✐❛ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ❍✉♥❣r✐❛ ❡ ❞♦ ■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇✉❞❛♣❡st❡✳ ❊ss❛ ✐❞❡✐❛ s❛❧✉t❛r ❢♦✐ ❞✐ss❡♠✐♥❛❞❛ ♣❡❧♦ r❡st♦ ❞❛ ❊✉r♦♣❛ ❡ ♣❛r❛ t♦❞♦ ♦ ♠✉♥❞♦✳

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❙✉♠ár✐♦ ✷

❉❡s❞❡ ✶✾✺✾✱ ❛❝♦♥t❡❝❡✱ ❛♥✉❛❧♠❡♥t❡✱ s❡♠♣r❡ ❡♠ ✉♠ ♣❛ís ❞✐❢❡r❡♥t❡✱ ❛ ❖❧✐♠♣í❛❞❛ ■♥t❡r✲ ♥❛❝✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ✭■♥t❡r♥❛t✐♦♥❛❧ ▼❛t❤❡♠❛t✐❝❛❧ ❖❧②♠♣✐❛❞ ✲ ■▼❖✮✳ ❖ ❇r❛s✐❧ t❡♠ ♣❛rt✐❝✐♣❛❞♦ ❞❛ ■▼❖ ❡ ♦❜t✐❞♦✱ ❛tr❛✈és ❞❡ s❡✉s ❥♦✈❡♥s✱ ❞✐✈❡rs❛s ♠❡❞❛❧❤❛s ❞❡ ♦✉r♦✳ ❊①✐st❡ t❛♠❜é♠✱ ❞❡s❞❡ ✶✾✽✺✱ ♣❛tr♦❝✐♥❛❞❛ ♣❡❧❛ ❖r❣❛♥✐③❛çã♦ ❞♦s ❊st❛❞♦s ■❜❡r♦✲❆♠❡r✐❝❛♥♦s ♣❛r❛ ❛ ❊❞✉❝❛çã♦✱ ❈✐ê♥❝✐❛ ❡ ❈✉❧t✉r❛✱ ❛ ❖❧✐♠♣í❛❞❛ ■❜❡r♦✲❆♠❡r✐❝❛♥❛ ❞❡ ▼❛t❡♠át✐❝❛✳ ◆❡ss❛ ❖❧✐♠♣í❛❞❛✱ ♦ ❇r❛s✐❧ ❥á ❝♦♥q✉✐st♦✉✱ ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✱ ♠❡❞❛❧❤❛s ❞❡ ♦✉r♦✱ ♣r❛t❛ ❡ ❜r♦♥③❡✳ ❆❝♦♥t❡❝❡ t❛♠❜é♠✱ ❛♥✉❛❧♠❡♥t❡✱ s❡♠♣r❡ ❡♠ ✉♠ ♣❛ís ❞✐❢❡r❡♥t❡✱ ❛ ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❈♦♥❡ ❙✉❧✱ ❡♥✈♦❧✈❡♥❞♦ ❡st✉❞❛♥t❡s ❞♦ ❇r❛s✐❧✱ ❆r❣❡♥t✐♥❛✱ ❇♦❧í✈✐❛✱ ❈❤✐❧❡✱ ❊q✉❛❞♦r✱ P❛r❛✲ ❣✉❛✐✱ P❡r✉ ❡ ❯r✉❣✉❛✐✳ ◆♦ ❇r❛s✐❧✱ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✕ ❙❇▼ ✲ ♣r♦♠♦✈❡✱ ❛♥✉❛❧♠❡♥t❡✱ ❞❡s❞❡ ✶✾✼✾✱ ❛ ❖❧✐♠♣í❛❞❛ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❖❇▼✮ ❡✱ ❛tr❛✈és ❞❛ ❈♦✲ ♠✐ssã♦ ❇r❛s✐❧❡✐r❛ ❞❡ ❖❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦♦r❞❡♥❛ ❛ ♣❛rt✐❝✐♣❛çã♦ ❞❡ ❡st✉❞❛♥t❡s ❜r❛s✐❧❡✐r♦s ❡♠ ❝♦♠♣❡t✐çõ❡s ✐♥t❡r♥❛❝✐♦♥❛✐s✳ P❡♥s❛♥❞♦ ♥❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s ❞❡ t♦❞♦ ♦ P❛ís✱ ♦ ▼❊❈ ✭▼✐♥✐stér✐♦ ❞❛ ❊❞✉❝❛çã♦✮✱ ❛ ❙❇▼ ❡ ♦ ■▼P❆ ✭■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✮ ❡stã♦ ♣r♦♠♦✈❡♥❞♦ ❡ ♦r❣❛♥✐③❛♥❞♦✱ ❞❡s❞❡ ✷✵✵✺✱ ❛ ❖❇▼❊P ✭❖❧✐♠♣í❛❞❛ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛s ❊s❝♦❧❛s Pú❜❧✐❝❛s✮✳ ❆ ❖❇▼❊P ✈❡♠ ❝r❡s❝❡♥❞♦ ❛ ❝❛❞❛ ❛♥♦✱ ❝r✐❛♥❞♦ ✉♠ ❛♠❜✐❡♥t❡ ❡st✐♠✉❧❛♥t❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡♥tr❡ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞❡ t♦❞♦ ♦ ♣❛ís✳ ❊♠ ✷✵✶✷✱ ❝❡r❝❛ ❞❡ ✶✾✱✶ ♠✐❧❤õ❡s ❞❡ ❛❧✉♥♦s s❡ ✐♥s❝r❡✈❡r❛♠ ♥❛ ❝♦♠♣❡t✐çã♦ ❡ ✾✾✱✹✪ ❞♦s ♠✉♥✐❝í♣✐♦s ❜r❛s✐❧❡✐r♦s ❡st✐✈❡r❛♠ r❡♣r❡s❡♥t❛❞♦s✳

❖s s✉❝❡ss✐✈♦s r❡❝♦r❞❡s ❞❡ ♣❛rt✐❝✐♣❛çã♦ ❢❛③❡♠ ❞❛ ❖❇▼❊P ❛ ♠❛✐♦r ❖❧✐♠♣í❛❞❛ ❞❡ ▼❛t❡✲ ♠át✐❝❛ ❞♦ ♠✉♥❞♦✳ ❉❡s❞❡ ♦ ✐♥í❝✐♦ ❞❛ ❖❇▼❊P✱ ❞✉❛s ❡s❝♦❧❛s ♣✐❛✉✐❡♥s❡s ✈ê♠ s❡ ❞❡st❛❝❛♥❞♦✿ ❆ ❯♥✐❞❛❞❡ ❊s❝♦❧❛r ❊st❛❞✉❛❧ ❊♥s✐♥♦ ▼é❞✐♦ ❆✉❣✉st✐♥❤♦ ❇r❛♥❞ã♦ ❡ ❛ ❯♥✐❞❛❞❡ ❊s❝♦❧❛r ▼✉♥✐❝✐♣❛❧ ❚❡♦tô♥✐♦✱ ❛♠❜❛s ❧♦❝❛❧✐③❛❞❛s ❡♠ ❈♦❝❛❧ ❞♦s ❆❧✈❡s✳ ❊♠ ✷✵✶✵✱ ❡st✉❞❛♥t❡s ❞❛s ❞✉❛s ❡s❝♦❧❛s ❝♦♥q✉✐st❛r❛♠ q✉❛tr♦ ♠❡❞❛❧❤❛s ❞❡ ♦✉r♦✱ três ❞❡ ♣r❛t❛✱ ❝✐♥❝♦ ❞❡ ❜r♦♥③❡ ❡ ✶✷ ♠❡♥çõ❡s ❤♦♥r♦s❛s✳ ❊♠ ✷✵✶✶✱ ❢♦r❛♠ ✷✹ ♣r❡♠✐❛çõ❡s✱ s❡♥❞♦✿ ✶✵ ♠❡❞❛❧❤❛s ❡ ✶✹ ♠❡♥çõ❡s ❤♦♥r♦s❛s✳

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❙✉♠ár✐♦ ✸

◆❛s ♣r♦✈❛s ❞❡ ❖❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✱ ❤á ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✲ ✈❡♥❞♦ ❞❡s✐❣✉❛❧❞❛❞❡s ❡✱ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❞❡ss❛ ♥❛t✉r❡③❛✱ ❛♣r❡s❡♥t❛r❡✲ ♠♦s ❛ ❢❡rr❛♠❡♥t❛ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ❞❡ ✉♠❛ ❱❛r✐á✈❡❧ ❘❡❛❧✳

❋✉♥çõ❡s ❈♦♥✈❡①❛s ❞❡ ✉♠❛ ❱❛r✐á✈❡❧ ❘❡❛❧ ❢♦r♠❛ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝❧❛ss❡ ❞❡ ❢✉♥çõ❡s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❆♥á❧✐s❡ ❘❡❛❧✳ ❊❧❛s sã♦ ♠✉✐t♦ ✉t✐❧✐③❛❞❛s ❡♠ ❖t✐♠✐③❛çã♦✱ ❜❡♠ ❝♦♠♦ ❡♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✳ ❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❝♦♥✈❡①❛s é q✉❡ ♦ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ✭q✉❛♥❞♦ ❡①✐st❡✮ é ❣❧♦❜❛❧✱ ♣♦✐s ✉♠❛ ❢✉♥çã♦ q✉❛❧q✉❡r ♣♦❞❡ t❡r ✈ár✐♦s ♣♦♥t♦s ❞❡ ♠í♥✐♠♦s ❧♦❝❛✐s✱ s❡♥❞♦ q✉❡ ❡❧❡s ♥❡♠ s❡♠♣r❡ sã♦ ♣♦♥t♦s ❞❡ ♠í♥✐♠♦ ❣❧♦❜❛✐s✳ ❆ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❡ ❞❡✜♥❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ♣❛r❛ ♠❛✐s ❞❡ ❞♦✐s ♣♦♥t♦s é ❝❤❛♠❛❞❛ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏❡♥s❡♥✱ ❣r❛ç❛s ❛♦ ❡♥❣❡♥❤❡✐r♦ ❞✐♥❛♠❛rq✉ês ❏♦❤❛♥ ▲✉❞✇✐❣ ❲✐❧❧✐❛♠ ❱❛❧❞❡♠❛r ❏❡♥s❡♥ ✭✶✽✺✾✲✶✾✷✺✮✳

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❈❛♣ít✉❧♦ ✶

❋✉♥çõ❡s ❈♦♥✈❡①❛s

❚r❛❜❛❧❤❛r❡♠♦s ❛s ❢✉♥çõ❡s ❝♦♥✈❡①❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❢✉♥çõ❡s r❡❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s✱ ♣r♦♣♦s✐çõ❡s ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛✲ çõ❡s s✐❣♥✐✜❝❛t✐✈❛s✳

❖ ❛ss✉♥t♦ ❛q✉✐ ❛❜♦r❞❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✽❪✳

✶✳✶ ❋✉♥çõ❡s ❈♦♥✈❡①❛s

❉❡✜♥✐çã♦ ✶✳ ❙❡❥❛♠−_∞6α < β6_∞✳ ❯♠❛ ❢✉♥çã♦ f: (α✱β)_⊂_R_→_R é ❞✐t❛ ❝♦♥✈❡①❛ s❡ t✐✈❡r ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❉❛❞♦s ❞♦✐s ♣♦♥t♦sA❡ B ♥♦ ❣rá✜❝♦ ❞❡f✱ ❛ ❝♦r❞❛ q✉❡ ✉♥❡

❡st❡s ❞♦✐s ♣♦♥t♦s ❡stá s❡♠♣r❡ ❛❝✐♠❛ ❞♦ ❣rá✜❝♦ ❞❡ f✳

❉❛❞♦sa6x 6b❡♠ (α✱β)✱ ❝❤❛♠❛♥❞♦ λ= x−a

b−a✱ t❡♠♦s ✵6λ6✶ ❡✿ x =a+ (x−a) =a+λ(b−a) = (✶−λ)a+λb✳

❋✐❣✉r❛ ✶✳✶✿

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❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✺

❖ s❡❣♠❡♥t♦ q✉❡ ♣❛ss❛ ♣♦rA ❡ Bt❡♠ ❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛✿

y=f(a) + f(b) −f(a)

b−a (x−a) =f(a) +λ(f(b) −f(a)) = (✶−λ)f(a) +λf(b)✱ ✵6λ6✶✳

❖✉ s❡❥❛✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❛ ❢✉♥çã♦✿

λ_7→((✶−λ)a+λb✱(✶−λ)f(a) +λf(b))✱ ✵6λ6✶✱

é ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❡♠R✷✳ ❆ss✐♠✱ ♦s ♣♦♥t♦sC ❡ D❞❛ ❋✐❣✉r❛ ✶✳✶ tê♠ ❝♦♦r❞❡♥❛❞❛s✿

C= ((✶−λ)a+λb✱f((✶−λ)a+λb)) ❡ D= ((✶−λ)a+λb✱(✶−λ)f(a) +λf(b))✳

❆ ❢✉♥çã♦ f é ❝♦♥✈❡①❛ q✉❛♥❞♦ ♦ ♣♦♥t♦ D ❡stá s❡♠♣r❡ ❛❝✐♠❛ ❞❡ C✳ ■st♦ s❡ ❡①♣r❡ss❛

❝♦♠♦✿

f((✶−λ)a+λb)6(✶−λ)f(a) +λf(b)❀_∀a✱b_∈(α✱β) ❡ ✵6λ6✶✳

P♦rt❛♥t♦✱ f é ❝♦♥✈❡①❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✿

f((✶−λ)a+λb)6(✶−λ)f(a) +λf(b)❀_∀a✱b_∈(α✱β) ❡ ✵6λ6✶✳ ✭✶✳✶✮

❉❡✜♥✐çã♦ ✷✳ ❙❡❥❛♠−_∞6α < β6∞✳ ❯♠❛ ❢✉♥çã♦ f: (α✱β)_⊂_R_→_R é ❞✐t❛ ❝ô♥❝❛✈❛ s❡ t✐✈❡r ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❉❛❞♦s ❞♦✐s ♣♦♥t♦sA❡ B ♥♦ ❣rá✜❝♦ ❞❡f✱ ❛ ❝♦r❞❛ q✉❡ ✉♥❡

❡st❡s ❞♦✐s ♣♦♥t♦s ❡stá s❡♠♣r❡ ❛❜❛✐①♦ ❞♦ ❣rá✜❝♦ ❞❡ f✳ ■st♦ s❡ ❡①♣r❡ss❛ ❝♦♠♦✿

f((✶−λ)a+λb)>(✶−λ)f(a) +λf(b)❀_∀a✱b_∈(α✱β) ❡ ✵6λ6✶✳ ✭✶✳✷✮

❊①❡♠♣❧♦ ✶✳ ❆ ❢✉♥çã♦ ♠♦❞✉❧❛rf(x) =|x| é ❝♦♥✈❡①❛✳

❙♦❧✉çã♦✿ ❙❡❥❛♠ a✱b_∈R✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ λ_∈[✵✱ ✶]✱ ✈❛❧❡✿

f((✶−λ)a+λb) = |(✶−λ)a+λb|

6 |(✶−λ)a|+|λb|

= (✶−λ)|a|+λ|b|

= (✶−λ)f(a) +λf(b)✳

P♦rt❛♥t♦✱ ♣♦r ✭✶✳✶✮ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❢✉♥çã♦ ♠♦❞✉❧❛r é ❝♦♥✈❡①❛✳

(16)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✻

❊①❡♠♣❧♦ ✷✳ ❯♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ f(x) =ax✷+bx+c✱ ❝♦♠ a > ✵ é ❝♦♥✈❡①❛✳

❙♦❧✉çã♦✿ ❙❡❥❛♠ a′✱b′ _∈ R ❡ s✉♣♦♥❤❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ a′ < b′✳

❊♥tã♦✱ ♣❛r❛ t♦❞♦ λ_∈[✵✱ ✶]✱ ✈❛❧❡✿

f((✶−λ)a′+λb′) =a((✶−λ)a′+λb′)✷+b((✶−λ)a′+λb′) +c

=a[(✶−λ)✷a′✷ +✷λ(✶−λ)a′b′+λ✷b′✷] + (✶−λ)ba′+λbb′+c

=a[(✶−λ)✷a′✷ +λ(✶−λ)✷a′b′+λ✷b′✷] + (✶−λ)ba′+λbb′+c

6(✶−λ)✷aa′✷+λ(✶−λ)a(a′✷+b′✷) +λ✷ab′✷+ (✶−λ)ba′+λbb′

+ (✶−λ)c+λc

= (✶−λ)[(✶−λ)aa′✷+λa(a′✷+b′✷) +ba′+c] +λ✷ab′✷+λbb′+λc

= (✶−λ)(aa′✷+ba′+c) + (✶−λ)λab′✷+λ✷ab′✷+λbb′+λc

= (✶−λ)(aa′✷+ba′+c) +λab′✷+λbb′+λc

= (✶−λ)(aa′✷+ba′+c) +λ(ab′✷+bb′+c) = (✶−λ)f(a′) +λf(b′)✳

❖❜s❡r✈❛çã♦ ✷✳ ◆❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❡ ❛♣❛r❡❝❡ ♥❛ s♦❧✉çã♦✱ ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ a s❡r ♣♦s✐t✐✈♦

❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ a′_b′₆ a′✷+b′✷

✷ ✳

❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❝♦♥✈❡①❛s é q✉❡ ♦ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ✭q✉❛♥❞♦ ❡①✐st❡✮ é ❣❧♦❜❛❧✳

❘❡s♦❧✈❡r❡♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛çã♦ ❝♦♥✈❡①❛✱♣❛r❛ ✐st♦✱ ❢♦r♠❛❧✐③❛r❡♠♦s ❛ ❞❡✜✲ ♥✐çã♦ ❞❡ ♠í♥✐♠♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳

❉❡✜♥✐çã♦ ✸✳ ❙❡❥❛ f:I_⊂R→R ✉♠❛ ❢✉♥çã♦ q✉❛❧q✉❡r✳ ❯♠ ♣♦♥t♦ c_∈I é✿

✭✐✮ ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f s❡ ❡①✐st❡ r >✵ t❛❧ q✉❡ ✿ |x−c|< r ✐♠♣❧✐❝❛ f(c) 6f(x)✱ ♣❛r❛ t♦❞♦ x_∈(c−r✱c+r)❀

✭✐✐✮ ✉♠ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ f s❡ f(c)6f(x)✱ ♣❛r❛ t♦❞♦ x_∈I✳

Pr♦♣♦s✐çã♦ ✶✳ ❙❡❥❛♠ f:I _⊂R→R ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ❡ c ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f ❡♠ I✳ ❊♥tã♦ c é ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ f ❡♠ I✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡f ❛❞♠✐t❛ ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧c q✉❡ ♥ã♦ é ♠í♥✐♠♦ ❣❧♦❜❛❧✳

(17)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✼

❊s❝♦❧❤❛λs✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❞❛ ✉♥✐❞❛❞❡ t❛❧ q✉❡ λc+ (✶−λ)z_∈(c−r✱c+r)✳ P❡❧❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❡f✱ t❡♠♦s✿

f(λc+ (✶−λ)z)6λf(c) + (✶−λ)f(z)< f(c)✳

P♦✐s✱f(z)< f(c)✳ ▼❛s✱ ♣❡❧❛ ❝♦♥str✉çã♦✱λc+ (✶−λ)z_∈(c−r✱c+r)✳ ❈♦♠♦c✉♠ ♠í♥✐♠♦

❧♦❝❛❧✱ f(c)6f(λc+ (✶−λ)z)✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱c é ✉♠ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡f❡♠

I✳

❖ r❡s✉❧t❛❞♦ ❛❜❛✐①♦ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✭✶✳✶✮✳

Pr♦♣♦s✐çã♦ ✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏❡♥s❡♥✮✳ ❙❡❥❛♠ −_∞ 6 α < β 6 ∞✳ ❯♠❛ ❢✉♥✲

çã♦ f : (α✱β) _⊂ _R _→ _R é ❝♦♥✈❡①❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ _∀ n > ✷✱_∀ x✶✱x✷✱ ✳ ✳ ✳ ✱xn ∈ (α✱β)✱_∀ λ_✶✱λ_✷✱ ✳ ✳ ✳ ✱λn ∈[✵✱ ✶]✱ t❛✐s q✉❡✱ λ✶+λ✷+✳ ✳ ✳+λn =✶✱ t❡♠✲s❡✿

f( n X

i=✶

λixi)6 n X

i=✶

λif(xi)✳ ✭✶✳✸✮

❉❡♠♦♥str❛çã♦✳ (_⇐) ❙✉♣♦♥❞♦ ✈á❧✐❞❛ ❛ ❝♦♥❞✐çã♦ ✭✶✳✸✮ ❡ t♦♠❛♥❞♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ n = ✷✱ ❝♦♥❝❧✉✐✲s❡ q✉❡f é ❝♦♥✈❡①❛✳

(_⇒)❙✉♣♦♥❤❛♠♦s q✉❡fé ❝♦♥✈❡①❛✳ ❱❛♠♦s ♣r♦✈❛r ♣♦r ✐♥❞✉çã♦ s♦❜r❡n✱ q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦

✭✶✳✸✮ ♣❛r❛ t♦❞♦n>✷✳

✭✐✮ P❛r❛ n=✷ é ✈á❧✐❞♦ ♣♦r ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❝♦♥✈❡①❛✳

✭✐✐✮ ❍✐♣ót❡s❡ ❞❡ ■♥❞✉çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ ✭✶✳✸✮ ✈❛❧❡ ♣❛r❛ ✉♠ ❝❡rt♦ n >✷✳

✭✐✐✐✮ ❚❡s❡✿ ❱❛♠♦s ♠♦str❛r q✉❡ ✭✶✳✸✮ ✈❛❧❡ t❛♠❜é♠ ♣❛r❛ n+✶✳ ❖✉ s❡❥❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡✿

f(λ_✶x_✶+λ_✷x_✷+✳ ✳ ✳+λnxn+λn+_✶xn+_✶)6λ_✶f(x_✶) +λ_✷f(x_✷) +✳ ✳ ✳+λnf(xn) +λn+_✶f(xn+_✶)✳ ❙❡❥❛♠x_✶✱x_✷✱ ✳ ✳ ✳ ✱xn✱xn+✶ ∈(α✱β) ❡ λ✶✱λ✷✱ ✳ ✳ ✳ ✱λn✱λn+✶ >✵✱ ❝♦♠ λ✶+λ✷+✳ ✳ ✳+λn+

λn+_✶ =✶✳ ✶♦ _{❝❛s♦✿} _λ

n+✶ =✶✳

❊♥tã♦✱ λ_✶ = λ_✷ =✳ ✳ ✳ = λn =✵ ❡✱ ♥❡st❡ ❝❛s♦✱ ❛ ❝♦♥❞✐çã♦ ✭✶✳✸✮ ✈❛❧❡ tr✐✈✐❛❧♠❡♥t❡✱ ♣♦✐s s❡ r❡❞✉③ ❛f(xn+✶)6f(xn+✶)✳

✷♦ _{❝❛s♦✿} _λ

n+_✶ 6=✶✳ ❈❤❛♠❛♥❞♦

(18)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✽

❡ ♦❜s❡r✈❛♥❞♦ q✉❡λ_✶+λ_✷+✳ ✳ ✳+λn=✶−λn+✶✱ t❡♠♦s q✉❡✿

λ_✶

✶−λn+✶

+ λ✷ ✶−λn+✶

+✳ ✳ ✳+ λn ✶−λn+✶

=✶✳

P❡❧❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❡f ❡ ♣❡❧❛ ❍✐♣ót❡s❡ ❞❡ ■♥❞✉çã♦✱ t❡♠♦s✿

f(λ_✶x_✶+λ_✷x_✷+✳ ✳ ✳+λnxn+λn+✶xn+✶) = f((✶−λn+✶)y+λn+✶xn+✶)

6 (✶−λn+✶)f(y) +λn+✶f(xn+✶) = (✶−λn+✶)f

λ_✶x_✶+λ_✷x_✷+✳ ✳ ✳+λnxn ✶−λn+✶

+λn+✶f(xn+✶)

6 (✶−λn+✶)

λ_✶

✶−λn+✶

f(x_✶) + λ✷ ✶−λn+✶

f(x_✷) +✳ ✳ ✳+ λn ✶−λn+✶

f(xn)

+λn+✶f(xn+✶) = λ_✶f(x_✶) +λ_✷f(x_✷) +✳ ✳ ✳+λnf(xn) +λn+✶f(xn+✶)✳

❖❜s❡r✈❛çã♦ ✸✳ ◗✉❛♥❞♦ λ_✶ =λ_✷ =✳ ✳ ✳=λn= ✶

n✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏❡♥s❡♥ ♥♦s ❞✐③ q✉❡✿

f

x_✶+x_✷+✳✳✳+xn

n

6 f(x✶) +f(x✷) +✳✳✳+f(xn)

n ✳

✶✳✷ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ❉❡r✐✈á✈❡✐s

❙❡ ♥♦s r❡str✐♥❣✐r♠♦s à ❝❧❛ss❡ ❞❛s ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s✱ ✈❛♠♦s ♦❜t❡r ✉♠ ❝r✐tér✐♦ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠❛ ❢✉♥çã♦ é ❝♦♥✈❡①❛✳ ❯♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡r✐✈á✈❡❧✳ P♦r ❡①❡♠♣❧♦✱ t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ✭✻✮✱ ❛ ❢✉♥çã♦ f(x) = |x| ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = ✵ ❡ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✶ q✉❡ ❡❧❛ é ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✳

Pr♦♣♦s✐çã♦ ✸✳ ❙❡❥❛f:I_→_R✉♠❛ ❢✉♥çã♦ f❝♦♥tí♥✉❛ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦I✳

❙❡ f t❡♠ ♠í♥✐♠♦ ❧♦❝❛❧ ❡♠ x=c✱c_∈I ❡ fé ❞❡r✐✈á✈❡❧ ❡♠ c ❡♥tã♦ f′(c) =✵✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ft❡♥❤❛ ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧ ❡♠ x=c✳ ❈♦♠♦ fé ❞❡r✐✈á✈❡❧ ❡♠ c✱ ❡♥tã♦

❧✐♠ x→c−

f(x) −f(c)

x−c =x❧✐♠→c+

f(x) −f(c)

x−c =x❧✐♠→c

f(x) −f(c)

x−c =f

′₍_c₎_✳

❈♦♠♦ c é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✱ ❡①✐st❡ r > ✵ t❛❧ q✉❡ f(c) 6 f(x)✱ ♣❛r❛ t♦❞♦ x _∈

(19)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✾

❙❡x < c ❡♥tã♦ x−c <✵✱ ❡✱ ♣♦rt❛♥t♦✱ f(x) −f(c)

x−c 6✵ ♣❛r❛ x∈(c−r✱c+r)✱ ❧♦❣♦✿

❧✐♠ x→c−

f(x) −f(c)

x−c 6✵✳ ✭✶✳✹✮

P♦r ♦✉tr♦ ❧❛❞♦✱x > c❡♥tã♦ x−c >✵ ❡✱ ♣♦rt❛♥t♦✱ f(x) −f(c)

x−c >✵ ♣❛r❛x ∈(c−r✱c+r)✿

❧✐♠ x→c+

f(x) −f(c)

x−c >✵✳ ✭✶✳✺✮

❈♦♠♣❛r❛♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✭✶✳✹✮ ❡ ✭✶✳✺✮ ❡ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ q✉❡ sã♦ ♦ ♠❡s♠♦ ✈❛❧♦r✱ r❡s✉❧t❛✿

❧✐♠ x→c

f(x) −f(c)

x−c =f

′₍_c_{) =}_✵✳

❊①❡♠♣❧♦ ✸✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❛ ❢✉♥çã♦ f ❞❛❞❛ ♣♦r f(x) =✷x✷ +✶✳ ❙♦❧✉çã♦✿ ❱✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✷ q✉❡ ❢ é ❝♦♥✈❡①❛ ❡ ❝♦♠♦ f′(x) =✹x✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶ s❡

❢ ❛❞♠✐t❡ ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧✱ ❡ss❡ ♠í♥✐♠♦ é ❣❧♦❜❛❧✳ P♦rt❛♥t♦✱ ❝♦♠♦ fé ❞❡r✐✈á✈❡❧ ❡♠ x=✵ ❡ f′(✵) = ✵✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✱ ❡ss❡ ♠í♥✐♠♦ ♦❝♦rr❡ ❡♠ x = ✵✳ ❆ss✐♠✱ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞❡ f é f(✵) =✶✳

Pr♦♣♦s✐çã♦ ✹✳ ❙❡❥❛♠ A= (x✶✱y✶)✱ B= (u✱v) ❡ C= (x✸✱y✸) três ♣♦♥t♦s ❞✐st✐♥t♦s s♦❜r❡

♦ ❣rá✜❝♦ ❞❡ f: _R _→R✱ ♦♥❞❡ f é ❝♦♥✈❡①❛✱ ❝♦♠ x_✶ < u < x_✸✳ ❊♥tã♦ ❛s três ♣r♦♣r✐❡❞❛❞❡s

s❡❣✉✐♥t❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✭❛✮ ❖ ♣♦♥t♦ B ❡stá ❛❜❛✐①♦ ❞❛ ❝♦r❞❛ AC❀

✭❜✮ ❆ ✐♥❝❧✐♥❛çã♦ ❞❡ AB é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ✐♥❝❧✐♥❛çã♦ ❞❡ AC❀

✭❝✮ ❆ ✐♥❝❧✐♥❛çã♦ ❞❡ AC é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ✐♥❝❧✐♥❛çã♦ ❞❡ BC✳

(20)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✵

❉❡♠♦♥str❛çã♦✳ ✭❛✮⇒ ✭❜✮✿ ❆ ♣r♦♣r✐❡❞❛❞❡ ✭❛✮ q✉❡ ❞✐③❡r q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ u ♣♦r f ❡stá

❛❜❛✐①♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A ❡ C ✭❡ ♣❛ss❛ ♣♦r D✮✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❋✐❣✉r❛ ✶✳✷✱ ❛

❡q✉❛çã♦ ❞❡ss❛ r❡t❛ é ❞❛❞❛ ♣♦r✿

y−z= y✸−y✶

x_✸−x_✶(x−u)✳ ✭✶✳✻✮

❈♦♠♦ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A ❡ C ✭❡ ♣❛ss❛ ♣♦r D✮ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦

✭✶✳✻✮✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦ ♣♦♥t♦A= (x✶✱y✶)✱ ♦✉ s❡❥❛✿

y_✶−z = y✸−y✶

x_✸−x_✶(x✶−u)⇒z =y✶+

y_✸−y_✶

x_✸−x_✶(u−x✶)✳

❈♦♠ ✐ss♦ ❛ ✐♠❛❣❡♠ ❞❡ u ♣♦r f é ♠❡♥♦r q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ u ♣❡❧❛ r❡t❛ ❞❛❞❛ ♣♦r ✭✶✳✻✮✳

❆ss✐♠✱

v6z _⇒v6y_✶+ y✸−y✶

x_✸−x_✶(u−x✶)✳

❈♦♠♦u−x_✶ >✵✱ t❡♠♦s✿

v−y_✶ u−x_✶ 6

y_✸−y_✶ x_✸−x_✶✳

✭❜✮⇒ ✭❝✮✿

v−y✶

u−x_✶ 6

y✸−y✶

x_✸−x_✶✳

❈♦♠♦u−x_✶ >✵ ❡ x_✸−x_✶ >✵✱ ♣♦❞❡♠♦s ❡♥tã♦ r❡❡s❝r❡✈❡r ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛ss✐♠✿

(v−y_✶)(x_✸−x_✶)6(y_✸−y_✶)(u−x_✶)

⇒ vx_✸−vx_✶−y_✶x_✸+y_✶x_✶ 6y_✸u−y_✸x_✶−y_✶u+y_✶x_✶

⇒ vx_✸−vx_✶−y_✶x_✸ 6y_✸u−y_✸x_✶−y_✶u✳

❙♦♠❛♥❞♦ ❛❣♦r❛x_✸y_✸ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s✿ x_✸y_✸−y_✶x_✸ +uy_✶−uy_✸ 6x_✸y_✸−y_✸x_✶−vx_✸+vx_✶

⇒ x_✸(y_✸ −y_✶) −u(y_✸ −y_✶)6y_✸(x_✸−x_✶) −v(x_✸−x_✶)

⇒ (y_✸−y_✶)(x_✸−u)>(x_✸−x_✶)(y_✸−v)✳ ❈♦♠♦(x✸−u)>✵ ❡ (x✸−x✶)>✵✱ ❡♥tã♦✿

y_✸−y_✶ x_✸−x_✶ 6

y_✸−v x_✸−u✳

✭❝✮⇒ ✭❛✮✿

y✸−y✶

x_✸−x_✶ 6

y✸−v

(21)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✶

❈♦♠♦x_✸−u >✵ ❡ x_✸−x_✶ >✵✱ ♣♦❞❡♠♦s ❡♥tã♦ r❡❡s❝r❡✈❡r ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛ss✐♠✿

(y_✸−y_✶)(x_✸−u)6(y_✸−v)(x_✸−x_✶)

⇒ y_✸x_✸−uy_✸−y_✶x_✸+y_✶u6y_✸x_✸−y_✸x_✶−vx_✸+vx_✶

⇒ −uy_✸−y_✶x_✸+y_✶u6−y_✸x_✶−vx_✸+vx_✶

⇒ vx_✸−vx_✶−y_✶x_✸ 6uy_✸−y_✸x_✶−uy_✶✳

❙♦♠❛♥❞♦ ❛❣♦r❛x_✶y_✶ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s✿ x_✶y_✶+vx_✸−vx_✶−y_✶x_✸ 6x_✶y_✶+uy_✸−y_✸x_✶−uy_✶

⇒ vx_✸−vx_✶−y_✶x_✸+y_✶x_✶ 6uy_✸−uy_✶−x_✶y_✸+x_✶y_✶

⇒ v(x_✸−x_✶) −y_✶(x_✸−x_✶)6u(y_✸−y_✶) −x_✶(y_✸−y_✶)

⇒ (x_✸−x_✶)(v−y_✶)6(y_✸ −y_✶)(u−x_✶)✳ ❈♦♠♦(x_✸−x_✶)>✵✱ ❡♥tã♦✿

v6y_✶+y✸−y✶

x_✸−x_✶(u−x✶)✳

◆❛s ♣r♦♣♦s✐çõ❡s ✺ ❡ ✻✱ ✉s❛r❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❛ Pr♦♣♦s✐çã♦ ✹✳

Pr♦♣♦s✐çã♦ ✺✳ ❙❡❥❛ f : (α✱β) _⊂ _R _→ _R ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✳ ❉❛❞♦s q✉❛tr♦ ♣♦♥t♦s

x_✶ < x_✷ < x_✸ < x_✹ ❡♠ (α✱β)✱ ✈❛❧❡ f(x✷) −f(x✶)

x✷−x✶

6 f(x✹) −f(x✸)

x✹−x✸ ✳

(22)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✷

❉❡♠♦♥str❛çã♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❋✐❣✉r❛ ✶✳✸✱ t❡♠♦s q✉❡ ♦ ♣♦♥t♦ B ❡stá ❛❜❛✐①♦ ❞❛ ❝♦r❞❛ AC✳ P❡❧❛ Pr♦♣♦s✐❝ã♦ ✹✱ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ ABé ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡AC ❡ ❡st❛ é

♠❡♥♦r ♦✉ ✐❣✉❛❧ à ✐♥❝❧✐♥❛çã♦ ❞❡BC✳ ❖✉ s❡❥❛✿ f(x_✷) −f(x_✶)

x✷−x✶ 6

f(x_✸) −f(x_✶)

x✸ −x✶ 6

f(x_✸) −f(x_✷)

x✸ −x✷ ✳ ✭✶✳✼✮

▲♦❣♦✱ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ ABé ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ✐♥❝❧✐♥❛çã♦ ❞❡ BC✳

❖ ♣♦♥t♦ C ❡stá ❛❜❛✐①♦ ❞❛ ❝♦r❞❛ BD✳ ▲♦❣♦✱ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ BC é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à

✐♥❝❧✐♥❛çã♦ ❞❡BD❡ ❡st❛ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ✐♥❝❧✐♥❛çã♦ ❞❡ CD✳ ❖✉ s❡❥❛✿ f(x_✸) −f(x_✷)

x_✸−x_✷ 6

f(x_✹) −f(x_✷)

x_✹ −x_✷ 6

f(x_✹) −f(x_✸)

x_✹ −x_✸ ✳ ✭✶✳✽✮

P♦rt❛♥t♦ ❞❡ ✭✶✳✼✮ ❡ ✭✶✳✽✮ ❝♦♥❝❧✉í♠♦s q✉❡✿

f(x_✷) −f(x_✶)

x_✷−x_✶ 6

f(x_✹) −f(x_✸)

x_✹−x_✸ ✳

Pr♦♣♦s✐çã♦ ✻✳ ❙❡❥❛ f : (α✱β) _⊂ _R _→ _R ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✳ ❊♥tã♦ f é ❝♦♥tí♥✉❛ ❡♠

(α✱β)✳

❉❡♠♦♥str❛çã♦✳ ❉❛❞♦ x _∈ (α✱β)✱ ❡s❝♦❧❤❛ δ > ✵ t❛❧ q✉❡ [x −δ✱x+δ] _⊂ (α✱β)✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠ ♣♦♥t♦z_✶ _∈(x−δ✱x)✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✹ ♣❛r❛ ♦s ♣♦♥t♦s x−δ✱z_✶✱x✱

t❡♠♦s✿

f(z_✶) −f(x−δ)

z✶− (x−δ)

6 f(x) −f(x−δ)

δ 6

f(x) −f(z_✶)

x−z✶ ✳ ✭✶✳✾✮

❆❣♦r❛✱ ❛♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✹ ♣❛r❛ ♦s ♣♦♥t♦s z_✶✱x✱x+δ✱ t❡♠♦s✿ f(x) −f(z_✶)

x−z✶

6 f(x+δ) −f(z✶)

(x+δ) −z✶

6 f(x+δ) −f(x)

δ ✳ ✭✶✳✶✵✮

▲♦❣♦✱ ❞❡ ✭✶✳✾✮ ❡ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s✿

f(x) −f(x−δ)

δ 6

f(x) −f(z_✶)

x−z_✶ 6

f(x+δ) −f(x)

δ ✳ ✭✶✳✶✶✮

❙❡❥❛ ❛❣♦r❛ ✉♠ ♣♦♥t♦z_✷ _∈(x✱x+δ)✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✹ ♣❛r❛ ♦s ♣♦♥t♦s x−δ✱x✱z_✷✱

t❡♠♦s✿

f(x) −f(x−δ)

δ 6

f(z_✷) −f(x−δ)

z_✷ − (x−δ) 6

f(z_✷) −f(x)

z_✷−x ✳ ✭✶✳✶✷✮

❆❣♦r❛✱ ❛♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✹ ♣❛r❛ ♦s ♣♦♥t♦s x✱z_✷✱x+δ✱ t❡♠♦s✿ f(z_✷) −f(x)

z_✷−x 6

f(x+δ) −f(x)

δ 6

f(x+δ) −f(z_✷)

(23)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✸

▲♦❣♦✱ ❞❡ ✭✶✳✶✷✮❡ ✭✶✳✶✸✮✱ ♦❜t❡♠♦s✿

f(x) −f(x−δ)

δ 6

f(z_✷) −f(x)

z_✷ −x 6

f(x+δ) −f(x)

δ ✳ ✭✶✳✶✹✮

❊♥tã♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❞❡ ✭✶✳✶✶✮ ❡ ✭✶✳✶✹✮✱ q✉❡ s❡✱z_∈[x−δ✱x+δ]✱

f(x) −f(x−δ)

δ 6

f(x) −f(z)

x−z

6 f(x+δ) −f(x)

δ ✳ ✭✶✳✶✺✮

❙❡❥❛✿

L= f(x+δ) −f(x)

δ ✳

P♦rt❛♥t♦✱ t❡♠♦s q✉❡✿

|f(x) −f(z)|6L|x−z|✱

♣❛r❛ q✉❛❧q✉❡r z_∈[x−δ✱x+δ]✳ P❛ss❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ z_→x✱ ♦❜t❡♠♦s✿

❧✐♠

z→x|f(x) −f(z)|=✵✱

♦ q✉❡ ♣r♦✈❛ q✉❡ ❢ é ❝♦♥tí♥✉❛✳

❖❜s❡r✈❛çã♦ ✹✳ ❱❡r ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥❛ ❘❡❢❡rê♥❝✐❛ ✻✳

▼♦str❛r❡♠♦s ♥❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ é ❝♦♥✈❡①❛✳

Pr♦♣♦s✐çã♦ ✼✳ ❙❡❥❛ f: (α✱β) _⊂_R _→_R ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧✳ ❊♥tã♦ f é ❝♦♥✈❡①❛ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ f′ é ❝r❡s❝❡♥t❡ ✭✐st♦ é✿ x_✶ < x_✷ _⇒f′(x_✶)6f′(x_✷)✮✳

❉❡♠♦♥str❛çã♦✳ (_⇒) ❙✉♣♦♥❤❛♠♦s q✉❡ f ❝♦♥✈❡①❛ ❡ ❞❡r✐✈á✈❡❧✳ ❱❛♠♦s ♠♦str❛r q✉❡ f′ _é

❝r❡s❝❡♥t❡✳ ❙❡❥❛♠x✶ < x✷ < x✸ < x✹ ♣♦♥t♦s ❞❡ (α✱β)✳

P❡❧❛ Pr♦♣♦s✐çã♦ ✺✱ t❡♠✲s❡✿

f(x_✷) −f(x_✶)

x_✷−x_✶ 6

f(x_✹) −f(x_✸)

x_✹−x_✸ ✳

❋❛③❡♥❞♦x✷ →x✶ ❡ x✹ →x✸✱ ♦❜t❡♠♦s f′(x✶)6f′(x✸)✳ P♦rt❛♥t♦✱ f′ é ❝r❡s❝❡♥t❡✳

(_⇐) ❙✉♣♦♥❤❛♠♦s q✉❡ f′ _{s❡❥❛ ❝r❡s❝❡♥t❡✳ ❱❛♠♦s ♣r♦✈❛r q✉❡} _f _{é ❝♦♥✈❡①❛✳ ❙❡❥❛♠} _x

✶ < x✷

❡♠ (α✱β) ❡ ✵ < λ <✶ ✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡✿

f((✶−λ)x_✶+λx_✷)6(✶−λ)f(x_✶) +λf(x_✷)✳ ❋❛③❡♥❞♦µ=✶−λ✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡✿

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❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✹

❚❡♠♦s✿

µf(x_✶) +λf(x_✷) −f(µx_✶+λf(x_✷) = µf(x_✶) +λf(x_✷) − (λ+µ)f(µx_✶+λx_✷)

= µ[f(x_✶) −f(µx_✶+λx_✷)] +λ[f(x_✷) −f(µx_✶+λx_✷)]✳ ❙❡♥❞♦f✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ (α✱β)✱ ❡♥tã♦fé ❝♦♥tí♥✉❛ ❡♠(α✱β)✱ ♣♦rt❛♥t♦✱ ❝♦♥tí♥✉❛ ♥♦s ✐♥t❡r✈❛❧♦s(x_✶✱µx_✶+λx_✷)❡(µx_✶+λx_✷✱x_✷)✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♥❡ss❡s ✐♥t❡r✈❛❧♦s✳ ❖✉ s❡❥❛✱ ❡①✐st❡c_∈(x_✶✱µx_✶+λx_✷)❡d_∈(µx_✶+λx_✷✱x_✷)✱ t❛❧ q✉❡✿

f(µx_✶+λx_✷) −f(x_✶) =f′(c)(µx_✶+λx_✷−x_✶) =λ(x_✷−x_✶)f′(c)✳ ✭✶✳✶✻✮

f(x✷) −f(µx✶+λx✷) =f′(d)(x✷ − (µx✶+λx✷)) = µ(x✷−x✶)f′(d)✳ ✭✶✳✶✼✮

❖❜s❡r✈❡ q✉❡ c < d ❡ ❝♦♠♦f′ _{é ❝r❡s❝❡♥t❡✱ t❡♠♦s q✉❡} _f′₍_c₎₆_f′₍_d₎_✳

❉❡ ✭✶✳✶✻✮ ❡ ✭✶✳✶✼✮ ❝♦♥❝❧✉í♠♦s q✉❡✿

f(µx_✶+λx_✷) −f(x_✶)

λ 6

f(x_✷) −f(µx_✶+λx_✷)

µ

⇒ µ[f(µx_✶+λx_✷) −f(x_✶)]6λ[f(x_✷) −f(µx_✶+λx_✷)]

⇒ µf(x_✶) +λf(x_✷) − (λ+µ)f(µx_✶+λx_✷)>✵✳

❈♦♠♦λ+µ=✶✱ t❡♠♦s✿

µf(x_✶) +λf(x_✷) −f(µx_✶ +λx_✷)>✵✳

❖✉ s❡❥❛✿

f(µx_✶+λx_✷)6µf(x_✶) +λf(x_✷)✳

❙❡ f é ❝♦♥tí♥✉❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ I ❡ f′(x) > ✵ ♣❛r❛ t♦❞♦ x ✐♥t❡r✐♦r ❛ I✱ ❡♥tã♦ f é

❝r❡s❝❡♥t❡ ❡♠I ✭✈❡r r❡❢❡rê♥❝✐❛ ✻✮✳ P♦rt❛♥t♦✱ ♦ ❈♦r♦❧ár✐♦ ✶ ❛❜❛✐①♦✱ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦

Pr♦♣♦s✐çã♦ ✼✳

❈♦r♦❧ár✐♦ ✶✳ ❙❡ f : (α✱β) _→ _R é ❞✉❛s ✈❡③❡s ❞❡r✐✈á✈❡❧✱ ❡♥tã♦ f é ❝♦♥✈❡①❛ s❡ f′′₍_x₎ _>

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❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✺

Pr♦♣♦s✐çã♦ ✽✳ ❙❡❥❛ f✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (α✱β) ✳ ❙❡ f é ✉♠❛

❢✉♥çã♦ ❝♦♥✈❡①❛✱ ❞❛❞♦x_✶ _∈(α✱β)✱ ❡♥tã♦✿

f(x)>f(x_✶) +f′(x_✶)(x−x_✶)✱ ✭✶✳✶✽✮ ♣❛r❛ t♦❞♦x _∈(α✱β)✳

❉❡♠♦♥str❛çã♦✳ ❙❡x=x_✶ ❡♠ ✭✶✳✶✽✮ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡✿ f(x_✶) =f(x_✶)✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s s✉♣♦rx ₆=x_✶✳ ❈♦♠♦ ❢ é ❝♦♥✈❡①❛✱ ♣♦r ✭✶✳✶✮✱ t❡♠♦s✿

f((✶−λ)x_✶ +λx)6(✶−λ)f(x_✶) +λf(x)

⇒ f(x_✶+λ(x−x_✶))6f(x_✶) +λ(f(x) −f(x_✶))

⇒ f(x_✶+λ(x−x_✶)) −f(x_✶)6λ(f(x) −f(x_✶))

⇒ (x−x_✶)f(x✶+λ(x−x✶)) −f(x✶)

λ(x−x_✶) 6f(x) −f(x✶)

⇒ f(x)>f(x_✶) + (x−x_✶)f(x✶+λ(x−x✶)) −f(x✶)

λ(x−x✶) ✳

❋❛③❡♥❞♦h=λ(x−x_✶)✱ ❡ ❛♣❧✐❝❛♥❞♦ ❧✐♠✐t❡ q✉❛♥❞♦ h_→✵ ♦❜t❡♠♦s✿ f(x)>f(x_✶) +f′(x_✶)(x−x_✶)✳

❖❜s❡r✈❛çã♦ ✺✳ ◆❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✽✱ s✉♣♦♠♦s λ ₆=✵ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ✭✶✳✶✽✮✳

◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ♠♦str❛r❡♠♦s q✉❡ ❛ ✈♦❧t❛ ❞♦ ❈♦r♦❧ár✐♦ ✶ t❛♠❜é♠ é ✈á❧✐❞❛✳

Pr♦♣♦s✐çã♦ ✾✳ ❙❡❥❛ f : (α✱β) _→ _R ❞✉❛s ✈❡③❡s ❞❡r✐✈á✈❡❧✳ ❙❡ f′′(x) > ✵✱_∀x _∈ (α✱β)✱ ❡♥tã♦ fé ❝♦♥✈❡①❛✳

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❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✻

❉❡♠♦♥str❛çã♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✽✱ ❜❛st❛ ♣r♦✈❛r q✉❡ f(x) > f(x_✶) +f′₍_x

✶)(x−x✶)✱ ♣❛r❛

t♦❞♦x_∈(α✱β)✳ ✶♦ _{❝❛s♦✿} _{x > x}

✶✳

❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♥♦ ✐♥t❡r✈❛❧♦[x_✶✱x] t❡♠♦s q✉❡ ❡①✐st❡ ✉♠c_∈(x_✶✱x) t❛❧ q✉❡✿

f(x) −f(x✶) =f′(c)(x−x✶)✳ ✭✶✳✶✾✮

❈♦♠♦c_∈(α✱β)✱ ❡♥tã♦f′′(c)>✵✱ ❧♦❣♦ f′(x)é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡✱ ♣♦rt❛♥t♦ f′(x✶)6

f′₍_c₎_{✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❛ ✐♥❡q✉❛çã♦ ♣❡❧♦ ❢❛t♦r ♣♦s✐t✐✈♦} ₍_x₋_x

✶)✱ r❡s✉❧t❛✿

f′(c)(x−x_✶)>f′(x_✶)(x−x_✶)_⇒f(x_✶) +f′(c)(x−x_✶)>f(x_✶) +f′(x_✶)(x−x_✶)✳ ▼❛s✱ ♣♦r ✭✶✳✶✾✮✱f(x) =f(x_✶) +f′₍_c₎₍_x₋_x

✶)✱ ❧♦❣♦✿

f(x)>f(x_✶) +f′(x_✶)(x−x_✶)✳ ✷♦ _{❝❛s♦✿} _{x < x}

✶✳

❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♥♦ ✐♥t❡r✈❛❧♦[x✱x_✶]✱ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠c_∈(x✱x_✶) t❛❧ q✉❡✿

f(x_✶) −f(x) =f′(c)(x_✶−x)✳ ✭✶✳✷✵✮ ❈♦♠♦ c_∈ (α✱β)✱ ❡♥tã♦ f′′(c) >✵✱ ❧♦❣♦ f′(x) é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡ ♣♦rt❛♥t♦ f′(c) 6

f′₍_x

✶)✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❛ ✐♥❡q✉❛çã♦ ♣❡❧♦ ❢❛t♦r ♣♦s✐t✐✈♦ (x✶−x)✱ r❡s✉❧t❛✿

f′(c)(x_✶−x)6f′(x_✶)(x−x_✶)

⇒ −f′(c)(x−x_✶)>f′(x_✶)(x−x_✶)

⇒ f(x_✶) −f′(c)(x_✶−x)>f(x_✶) +f′(x_✶)(x−x_✶)✳ ▼❛s✱ ♣♦r ✭✶✳✷✵✮✱ f(x) =f(x_✶) −f′₍_c₎₍_x₋_x

✶)✱ ❧♦❣♦✿

f(x)>f(x_✶) +f′(x_✶)(x−x_✶)✳

❉❡✜♥✐çã♦ ✹✳ ❯♠❛ ❢✉♥çã♦f: (α✱β)_⊂_R_→_R é ❞✐t❛ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ q✉❛♥❞♦✱ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s A ❡ B❡♠ s❡✉ ❣rá✜❝♦✱ ❛❧é♠ ❞❛ ❝♦r❞❛ AB ❡st❛r ❛❝✐♠❛ ❞♦ ❣rá✜❝♦ ❞❡ f✱ ❡❧❛ t♦❝❛

♦ ❣rá✜❝♦ ❛♣❡♥❛s ♥♦s ♣♦♥t♦sA ❡ B✳ ❖✉ s❡❥❛✱f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ q✉❛♥❞♦✿

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❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✼

❊①❡♠♣❧♦ ✹✳ ❯♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ é ❝♦♥✈❡①❛✱ ♠❛s ♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

❖❜s❡r✈❛çã♦ ✻✳ ❖❧❤❛♥❞♦ ❛s Pr♦♣♦s✐çõ❡s ✼ ❡ ✾✱ t❡♠♦s q✉❡✿

✐✳ ❆ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b❀

✐✐✳ ❙❡ f é ❞❡r✐✈á✈❡❧✱ ❡♥tã♦ f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f′ é ❡str✐t❛♠❡♥t❡

❝r❡s❝❡♥t❡ (x_✶ < x_✷ _⇒f′₍_x

✶)< f′(x✷))❀

✐✐✐✳ ❙❡ fé ❞✉❛s ✈❡③❡s ❞❡r✐✈á✈❡❧ ❡ f′′(x)>✵✱_∀x_∈I✱ ❡♥tã♦ f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

❆ r❡❝í♣r♦❝❛ é ❢❛❧s❛✱ ♣♦✐s ❛ ❢✉♥çã♦ f(x) =x✹ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ ❡ f′′₍_✵_{) =}_✵✳

❊①❡♠♣❧♦ ✺ ✭❊①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛s✮✳

✭✐✮ f(x) =x✷n✱x_∈R ❡ n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦❀

✭✐✐✮ f(x) =xp✱x>✵ ❡ p >✶❀

✭✐✐✐✮ f(x) = ✶

(x+a)p✱x >−a ❡ p >✵❀ ✭✐✈✮ f(x) =ex✱_x _∈R❀

✭✈✮ f(x) =x❧♥x✱x >✵❀

✭✈✐✮ f(x) =❧♥(✶+ex₎_✱_x _∈_R❀

✭✈✐✐✮ f(x) =tgx✱ ✵6x < π

✷❀

✭✈✐✐✐✮ f(x) = ✶

✶+ex✱x >✵✳

❘❡s♦❧✈❡r❡♠♦s ♦s ✐t❡♥s ✭✈✮✱ ✭✈✐✮ ❡ ✭✈✐✐✐✮✱ ♣♦✐s ✐r❡♠♦s ✉t✐❧✐③❛r ❡ss❛s ❢✉♥çõ❡s ♥❛ ♣ró①✐♠❛ s❡çã♦✳

✭✈✮ f(x) =x❧♥x✱x >✵✳

❙♦❧✉çã♦✿ ❙❡♥❞♦ f(x) =x❧♥x ❝♦♠ x >✵✱ t❡♠♦s q✉❡ f′(x) =❧♥x+✶ ❡ f′′(x) = ✶

x✳

❈♦♠♦ f′′₍_x₎_>_✵✱_∀_x _∈_D

f✱ t❡♠♦s q✉❡ fé ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳ ✭✈✐✮ f(x) =❧♥(✶+ex₎_✱_x _∈_R✳

❙♦❧✉çã♦✿ ❙❡♥❞♦ f(x) =❧♥(✶+ex₎✱_x_∈R✳ ❚❡♠♦s✿

f′(x) = e x

✶+ex ❡ f

′′₍_x_{) =} ex(✶+ex) −e✷x

(✶+ex₎✷ =

ex

(✶+ex₎✷✳

(28)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✽

✭✈✐✐✐✮ f(x) = ✶

✶+ex✱x >✵✳ ❙♦❧✉çã♦✿ ❙❡♥❞♦ f(x) = ✶

✶+ex✳ ❚❡♠♦s✿

f′(x) = −e x

(✶+ex₎✷ ❡ f

′′₍_x_{) =} ex(ex−✶)

(ex₊✶₎✸ >✵✳ ❈♦♠♦ f′′₍_x₎_>_✵✱_∀_{x >}_{✵✱ t❡♠♦s q✉❡ ❢✉♥çã♦} _f _{é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳}

❖❜s❡r✈❛çã♦ ✼✳

✶✳ ❆ ❢✉♥çã♦ ❧✐♥❡❛r f(x) =ax+b ❝♦♠ x _∈_R é ❝♦♥✈❡①❛ ❡ t❛♠❜é♠ ❝ô♥❝❛✈❛✳

✷✳ ❆ s♦♠❛ ❞❡ ❞✉❛s ❢✉♥çõ❡s ❝♦♥✈❡①❛s ✭❝ô♥❝❛✈❛s✮ é ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ✭❝ô♥❝❛✈❛✮✳

✶✳✸ ❆♣❧✐❝❛çõ❡s

❊①❡♠♣❧♦ ✻✳ Pr♦✈❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r a✱b✱x ❡ y r❡❛✐s ♣♦s✐t✐✈♦s✿

(x+y)❧♥x+y

a+b 6x❧♥ x

a+y❧♥ y b✱

❝♦♠ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x

a = y b✳

❙♦❧✉çã♦✿ ❱✐♠♦s ♥♦ ✐t❡♠ ✭✈✮ ❞♦ ❊①❡♠♣❧♦ ✺ q✉❡ f(x) = x❧♥x✱x > ✵ é ❡str✐t❛♠❡♥t❡ ❝♦♥✲

✈❡①❛✳ ❉❛❞♦sa✱b >✵✱ t♦♠❛♥❞♦λ= b

a+b ❡ µ= a

a+b✱ t❡♠♦sλ+µ=✶ ❝♦♠ λ✱µ >

✵✳ P♦rt❛♥t♦✱

f

x+y a+b

=f

a a+b

x a+

b a+b

y b

6 a

a+bf

x

a

+ b

a+bf

y

b

✳

❆ss✐♠✱

x+y a+b❧♥

x+y a+b 6

a a+b

x a❧♥

x a +

b a+b

y b❧♥

y b

⇒ (x+y)❧♥

x+y a+b

6x❧♥ x

a+y❧♥ y b✳

❈♦♠♦ f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✱ ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞❛ ❖❜s❡r✈❛çã♦ ✻✱ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ x

a = y b✳

❊①❡♠♣❧♦ ✼✳ ❉❛❞♦s ♥ú♠❡r♦s r❡❛✐s α_✶✱α_✷✱ ✳ ✳ ✳ ✱αn >✵ t❛✐s q✉❡ α✶+α✷+✳ ✳ ✳+αn =✶ ✱ ❡♥tã♦ ♣❛r❛ t♦❞♦s r❡❛✐s a_✶✱a_✷✱ ✳ ✳ ✳ ✱an✱b_✶✱b_✷✱ ✳ ✳ ✳ ✱bn ♣♦s✐t✐✈♦s✱ t❡♠✲s❡✿

aα✶

✶ ✳ ✳ ✳ ✳ ✳aαnn +b α_✶

(29)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✶✾

❝♦♠ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ b✶

a_✶ =✳ ✳ ✳= bn

an✳

❙♦❧✉çã♦✿ ❱✐♠♦s ♥♦ ✐t❡♠ ✭✈✐✮ ❞♦ ❊①❡♠♣❧♦ ✺ q✉❡f(x) =❧♥(✶+ex)é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳ P♦rt❛♥t♦✿

f

α_✶❧♥b✶

a✶ +✳ ✳ ✳+αn❧♥

bn

an

6α_✶f

❧♥b✶

a✶

+✳ ✳ ✳+αnf

❧♥bn

an ⇒ f ❧♥ b_✶ a_✶ α_✶

+✳ ✳ ✳+ln

bn

an αn

6α✶❧♥

✶+e❧♥

b✶

a_✶

+✳ ✳ ✳+αn❧♥

✶+e❧♥ bn an ⇒ f ❧♥ b_✶ a_✶ α_✶ ✳ ✳ ✳ bn an αn

6α_✶❧♥

✶+ b✶

a_✶

+✳ ✳ ✳+αn❧♥

✶+bn

an

⇒ ❧♥

✶+e

❧♥ b_✶ a_✶ α_✶ ✳ ✳ ✳ bn an αn 6❧♥

a_✶+b_✶ a✶

α_✶ ✳ ✳ ✳

an+bn

an

αn

⇒ ❧♥

✶+ b

α_✶

✶ ✳ ✳ ✳ ✳ ✳bαnn

aα_✶✶✳ ✳ ✳ ✳ ✳aαn

n

6❧♥

(a_✶+b_✶)α✶✳ ✳ ✳ ✳ ✳₍_a

n+bn)αn

aα_✶✶✳ ✳ ✳ ✳ ✳aαn

n

✳

❉❛í s❡❣✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✶✮✳

❈♦♠♦ f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✱ ♦❝♦rr❡ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❧♥ b✶

a_✶ =✳ ✳ ✳ =❧♥ bn

an ✱

✐st♦ é✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ b✶

a_✶ =✳ ✳ ✳= bn

an

✳✭✈❡r ✐t❡♠ ✭✐✮ ❞❛ ❖❜s❡r✈❛çã♦ ✻✮✳

❊①❡♠♣❧♦ ✽✳ ❙❡a✱b✱c >✵✱ ❡♥tã♦ aa✳bb✳cc >

a+b+c

✸

a+b+c ✳ ❙♦❧✉çã♦✿ ❉❡✈❡♠♦s ♠♦str❛r q✉❡✿

❧♥(aa✳bb✳cc)>❧♥

a+b+c

✸

a+b+c

⇒ a❧♥a+b❧♥b+c❧♥c>(a+b+c)❧♥

a+b+c

✸

✳ ✭✶✳✷✷✮

❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ f(x) = x❧♥x✱x _∈ (✵✱_∞) ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ ✭✐t❡♠ ✭✈✮ ❞♦ ❊①❡♠♣❧♦ ✺✮✱ t❡♠♦s✿

f

a+b+c

✸

6 f(a) +f(b) +f(c)

✸

⇒

a+b+c

✸

❧♥

a+b+c

✸

6 a❧♥a+b❧♥b+c❧♥c

✸

⇒ (a+b+c)❧♥

a+b+c

✸

6a❧♥a+b❧♥b+c❧♥c✳

◗✉❡ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✷✮ q✉❡ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❊①❡♠♣❧♦ ✾✳ ❙❡a_✶✱a_✷✱ ✳ ✳ ✳ ✱an>✶✱ ❡♥tã♦ n

X

k=✶ ✶ ✶+ak

> n

✶+√n_a

(30)

❈❛♣ít✉❧♦ ✶✳ ❋✉♥çõ❡s ❈♦♥✈❡①❛s ✷✵

❙♦❧✉çã♦✿ ❙❡❥❛ f(x) = ✶

✶+ex✱x > ✵✳ ❆ ❢✉♥çã♦ f é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ ✭✐t❡♠ ✭✈✐✐✐✮ ❞♦ ❊①❡♠♣❧♦ ✺✮✳ ❆ss✐♠✱ ♣♦r ✭✶✳✸✮✱ t❡♠♦s✿

f

x_✶+x_✷+✳ ✳ ✳+xn

n

6 f(x✶) +f(x✷) +✳ ✳ ✳+f(xn)

n

⇒ ✶

✶+e

x_✶+x_✷+✳ ✳ ✳+xn

n

! 6

n P

k=✶ ✶ ✶+exk

n

⇒ n

✶+e

x_✶+x_✷+✳ ✳ ✳+xn

n

! 6

n X

k=_✶ ✶ ✶+exk✳

❚♦♠❛♥❞♦ xk=❧♥ak✱ ♦❜t❡♠♦s✿ n X

k=✶ ✶ ✶+ak

> n

✶+√n_a

(31)

❈❛♣ít✉❧♦ ✷

▼é❞✐❛s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❛s ♣r✐♥❝✐♣❛✐s ♠é❞✐❛s ❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡①✐st❡♥t❡s ❡♥tr❡ ❡❧❛s✳ ❉❡✜♥✐r❡♠♦s ❛ ♠é❞✐❛ ❞❛s ♣♦tê♥❝✐❛s ❞❡ ♦r❞❡♠ ♣ ✭♣✲♠é❞✐❛s✮✱ ✈❡rsõ❡s ♣♦♥❞❡r❛❞❛s ♣❛r❛ ♣✲♠é❞✐❛s ❡ ❛♦ ✜♥❛❧✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çõ❡s✳

❖ ❛ss✉♥t♦ ❛❜♦r❞❛❞♦ ♥❡ss❡ ❝❛♣ít✉❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✺❪✱ ❬✼❪ ❡ ❬✾❪✳

✷✳✶ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛ ✭▼❍✮

❉❡✜♥✐çã♦ ✺✳ ❙❡❥❛♠ a_✶✱a_✷✱ ✳ ✳ ✳ ✱an r❡❛✐s ♣♦s✐t✐✈♦s✱ ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞❡ a✶✱a✷✱ ✳ ✳ ✳ ✱an é ❞❛❞❛ ♣♦r✿

MH= _✶ n

a_✶ +

✶

a_✷ +✳ ✳ ✳+

✶

an ✳

❆ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ é ✉t✐❧✐③❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ r❡s✐stê♥❝✐❛ ♠é❞✐❛ ❞❡ r❡s✐st♦r❡s ❡♠ ♣❛r❛❧❡❧♦✳

✷✳✷ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ✭▼●✮

❉❡✜♥✐çã♦ ✻✳ ❙❡❥❛♠ a_✶✱a_✷✱ ✳ ✳ ✳ ✱an r❡❛✐s ♣♦s✐t✐✈♦s✱ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞❡ a✶✱a✷✱ ✳ ✳ ✳ ✱an é ❞❛❞❛ ♣♦r✿

MG= √n_a

✶a✷✳ ✳ ✳an✳

❆ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ é ✉t✐❧✐③❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ t❛①❛ ♠é❞✐❛ ❞❡ r❡t♦r♥♦✳