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(a) f′′(x) ≤ 0 for x ≥ 0 (b) Since t2/2 is convex we have t2/2 ≥ x2/2x(t−x) = xt−x2/2 This is the general inequality g(t) ≥ g(x)g′(x)(t−x), which holds for any differentiable convex function, applied to g(t) = t2/2 Another (easier?) way to establish t2/2 ≤ −x2/2xt is to note that t2/2x2/2−xt = (1/2)(x−t)2 ≥ 0 Now just move x2/2−xt to the other side.

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Simple and best practice solution for g=(xc)/x equation Check how easy it is, and learn it for the future Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework. H c Ղh m { x ^ ̏ڍ׏ ē Ă ܂ B" ́A Îғ Ɍ m F ̂ A 䎩 ̐ӔC ŁA 䗘 p B s T C g ߂ @ t I N V. L ̃C X g ^ { ݂̃C X g M D ق̂ڂ̃C X g C A j V , a A s Ȃ.

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΂ Ă ݁i { X ^ C X g j 쐪 Ɓi j Č i { 揈 j O c ؗR I q i H n C ^ j ʏ R i ۂ t j R c i 𕶉 j. (a) f′′(x) ≤ 0 for x ≥ 0 (b) Since t2/2 is convex we have t2/2 ≥ x2/2x(t−x) = xt−x2/2 This is the general inequality g(t) ≥ g(x)g′(x)(t−x), which holds for any differentiable convex function, applied to g(t) = t2/2 Another (easier?) way to establish t2/2 ≤ −x2/2xt is to note that t2/2x2/2−xt = (1/2)(x−t)2 ≥ 0 Now just move x2/2−xt to the other side. Luxman arro ASCS5 alexa Amazon Music ̐ڑ Luxman arro ASCS5 DLNA T o Ƃ̐ڑ ASCS5 ́A T C Y p b P W O l Ə\ ȉ ł 8,000 ~ ̉ i ̔ ȏ ̓f U C ⑽ @ \ ɔ ₳ Ă ł 傤 B ܂ A ^ ^ Ƃ ꂽ ̂ł͂ ͂ \ ȉ ̍L ͎ Ȃ 悤 ł B.

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Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

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Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

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Ae Aeœ E µe A Ae C Aºœae Ae Aº E Vicjuan S A Aººa A Cs A C

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