Let be twice continuously differentiable on with for every

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

Let ff be twice continuously differentiable on R\mathbb R with f′′(x)>0f''(x)\gt0 for every xx, and suppose f′f' takes every real value. ((a)) Explain why f′(x)=sf'(x)=s has exactly one solution x∗(s)x^\ast(s) for each real ss, why x∗x^\ast is a differentiable function of ss, and why sup⁡x(sx−f(x))\sup_x\bigl(sx-f(x)\bigr) is attained at x∗(s)x^\ast(s). ((b)) With g(s)=L[f](s)=s x∗(s)−f(x∗(s))g(s)=\mathcal L[f](s)=s\,x^\ast(s)-f(x^\ast(s)), show that g′(s)=x∗(s)g'(s)=x^\ast(s) and g′′(s)=1/f′′(x∗(s))g''(s)=1/f''(x^\ast(s)). ((c)) Show that L[g](x)=sup⁡s(xs−g(s))=f(x)\mathcal L[g](x)=\sup_s\bigl(xs-g(s)\bigr)=f(x) for every real xx. ((d)) Check parts (b) and (c) on f(x)=cosh⁡xf(x)=\cosh x: find g(s)g(s) and confirm that g′(s)=sinh⁡−1sg'(s)=\sinh^{-1}s.

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