4 (a) Show that the functions f(x)=eªª, ƒ(x)= -lnx, and f(x) = x² are all convex. (b) Show that if f is twice differentiable and convex, then the first order Taylor approximation of fis an underestimate of the function, that is, ƒ(a) + (x − a) ªf (a) ≤ f(x), dx for every a and x. (c) Show that if fhas the property in part (b), and if X is a random variable, then ƒ(E[X]) ≤ E[ƒ(X)].
4 (a) Show that the functions f(x)=eªª, ƒ(x)= -lnx, and f(x) = x² are all convex. (b) Show that if f is twice differentiable and convex, then the first order Taylor approximation of fis an underestimate of the function, that is, ƒ(a) + (x − a) ªf (a) ≤ f(x), dx for every a and x. (c) Show that if fhas the property in part (b), and if X is a random variable, then ƒ(E[X]) ≤ E[ƒ(X)].
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please no written by hand solution and no img
![Problem 3.
A twice differentiable real-valued function f of a single variable is called convex if its
second derivative (d²f / dx²)(a) is nonnegative for all in its domain of definition.
(a) Show that the functions f(x)=eº, ƒ(x) = − lnx, and f(x) = x² are all convex.
(b) Show that if f is twice differentiable and convex, then the first order Taylor
approximation of fis an underestimate of the function, that is,
df
ƒ(a)+(x− a) ª (a) ≤ f(x),
dx
for every a and x.
(c) Show that if fhas the property in part (b), and if X is a random variable, then
ƒ(E[X]) ≤ E[ƒ(X)].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad375ba6-f10d-4d2a-a092-7ac9bb4cdcc7%2F07935108-676e-4042-ab91-b77a312feb1c%2Fj3amemi_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3.
A twice differentiable real-valued function f of a single variable is called convex if its
second derivative (d²f / dx²)(a) is nonnegative for all in its domain of definition.
(a) Show that the functions f(x)=eº, ƒ(x) = − lnx, and f(x) = x² are all convex.
(b) Show that if f is twice differentiable and convex, then the first order Taylor
approximation of fis an underestimate of the function, that is,
df
ƒ(a)+(x− a) ª (a) ≤ f(x),
dx
for every a and x.
(c) Show that if fhas the property in part (b), and if X is a random variable, then
ƒ(E[X]) ≤ E[ƒ(X)].
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