Let $f\in {\mathcal{C}}^{2}$ (i.e, $f$ is differentiable twice and ${f}^{\prime},{f}^{\u2033}$ are continuous. Show that $f$ can be written as $f(x)=g(x)+h(x)$ where $g(x)$ is convex for any $x$ and $h(x)$ is concave for any $x$.

Adrian Brown
2022-11-20
Answered

Let $f\in {\mathcal{C}}^{2}$ (i.e, $f$ is differentiable twice and ${f}^{\prime},{f}^{\u2033}$ are continuous. Show that $f$ can be written as $f(x)=g(x)+h(x)$ where $g(x)$ is convex for any $x$ and $h(x)$ is concave for any $x$.

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Camden Stanton

Answered 2022-11-21
Author has **14** answers

Write ${f}^{\u2033}$ as the sum of a (not necessarily continuous) non-negative function and non-positive function. Then integrate twice.

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Positive constant divided by a concave function, how to convexify this constraint?

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Prove it:

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a concave function and $\theta >1$. Then

$${\theta}^{k}f(x)\ge f({\theta}^{k}x)$$

for all $k=1,2,\dots $

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a concave function and $\theta >1$. Then

$${\theta}^{k}f(x)\ge f({\theta}^{k}x)$$

for all $k=1,2,\dots $

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Does there exist a concave function $f:(0,\mathrm{\infty})\to (0,\mathrm{\infty})$ with the following properties?

$f$ is $r$-homogeneous for some $r>0$, i.e., $r>0$ for all $x>0$

$f$ is $r$-homogeneous for some $r>0$, i.e., $r>0$ for all $x>0$

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Is there a concave function that is positive and grows faster than this polynomial function as $x$ goes to $-\mathrm{\infty}$?

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Let $A\u228a{\mathbb{R}}^{n}$( A is compact). Consider $f:{\mathbb{R}}^{n}\to {\mathbb{R}}_{+}^{n}\cup \{0\}$ is continuous map. Suppose that for all $a\in A$ , $\alpha \mapsto f(\alpha )$ is concave positive function.

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Concave function - Generic proof

$$f(x)\le 2f\left(\frac{x}{2}\right)\le 3f\left(\frac{x}{3}\right)\le \cdots \le Nf\left(\frac{x}{N}\right)$$

$$f(x)\le 2f\left(\frac{x}{2}\right)\le 3f\left(\frac{x}{3}\right)\le \cdots \le Nf\left(\frac{x}{N}\right)$$

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Do $\underset{x\downarrow 0}{lim}f(x)$ and $\underset{x\uparrow 1}{lim}f(x)$ exist if $f$ is concave over $[0,1]$?