Define the function h(x)= _ 0 ^ x | t| ,dt for all real numbers .

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Define the function

h(x)=\int_{0}^{x} |\sin t|\,dt

for all real numbers

x

. (a) For

0< x< \pi

, determine

h'(x)

. (b) For

\pi< x< 2\pi

, determine

h'(x)

. (c) Analyze the differentiability of

h

x=\pi

. Justify your conclusion by examining one-sided derivatives.