Mathematics

Problem

Define

f(x)=|x|

for all real

x

. A function

G

is proposed:

G(x)=\tfrac12 x|x|+C,

where

C

is a constant. (a) Compute

G'(x)

separately for the cases

x>0

and

x<0

. Show all steps in each region. (b) Combine your work from part (a) to verify that

G'(x)=f(x)

for every

x\neq0

. Explain how this establishes that

G

is an antiderivative of

f

on each open interval

(-\infty,0)

and

(0,\infty)

. (c) Discuss briefly what happens at

x=0

. Does the conclusion in part (b) extend to all real numbers? Support your answer.

Intuitive explanation

The key insight is recognizing that $G(x)$ is designed to be an antiderivative of $f(x) = |x|$ . To verify this works, differentiate $G(x)$ piecewise—when $x > 0$ , you get $x$ , and when $x < 0$ , you get $-x$ , which perfectly matches $|x|$ in each region. The constant $C$ simply shifts the entire function vertically, so it doesn't affect the derivative. This is your chance to see how absolute value functions integrate and why piecewise definitions are essential.

Intuitive explanation

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