CPM Homework Help : APCALC Problem 3-94

Show that if $f^\prime$ is an even function and $f(0) = 0$ , then $f$ is odd. Demonstrate this fact with a graph.

Hint:

Sketch different examples of possible $f^\prime(x)$ functions that are both even and go through the origin. Then sketch their antiderivatives $f(x)$ .

Examples;

$f'(x)=x^{2}................f(x)=\frac{1}{3}x^{3}+C$

$f'(x)=x^{4}................f(x)=\frac{1}{5}x^{5}+C$

$f'(x)=\text{sin}x.............f(x)=-\text{cos}x+C$

Hint:

Make a conjecture about why this will work ONLY if the even derivative goes through the origin? For example: consider even function $f^\prime (x) = x^2 + 1$ .