Prove the results shown in graph. Definition 1. A function f : R → R is called conver if Jensen's inequality,f(0x + (1 – 0)y) 0 for every x E R.Theorem 2. If fi : R → R are conver functions for i = 1,2, .. ,n, then the pointwise marimum....f(x) = max{fi (x), f2(x),..., fn(x)}is also conver.

Answered: Theorem 1. If f : R → R is a C²…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Prove the results shown in graph.

Definition 1. A function f : R → R is called conver if Jensen's inequality,
f(0x + (1 – 0)y) <of(x)+(1 – 0) f (y)
holds for every x,y E R and every 0 E [0, 1].
Theorem 1. If f : R → R is a C² function, then f is conver if and only if f"(x) > 0 for every x E R.
Theorem 2. If fi : R → R are conver functions for i = 1,2, .
. ,n, then the pointwise marimum
....
f(x) = max{fi (x), f2(x),..., fn(x)}
is also conver.

Expert Solution

Step 1: Writing the given information

Let $f : R \to R$ is a $C^{2}$ function.

We need to prove that $f$ is a convex function if and only if $f^{″} (x) > 0$ , for every $x \in R$ .

Let $z = (1 - λ) x + λ y \in [x, y]$ .

Applying the mean value theorem to $f$ there exists $c_{1} \in (x, z)$ and $c_{2} \in (z, y)$ such that

$f open parentheses z close parentheses equals f open parentheses x close parentheses plus f apostrophe open parentheses c subscript 1 close parentheses open parentheses z minus x close parentheses$ and $f open parentheses y close parentheses equals f open parentheses z close parentheses plus f apostrophe open parentheses c subscript 2 close parentheses open parentheses y minus z close parentheses$ .