Use derivatives to sketch the graph of a function

by RoRi

September 30, 2015

Let

$f(x) = \sin^2 x.$

Find all points such that $f'(x) = 0$ ;
Determine the intervals on which $f$ is monotonic by examining the sign of $f'$ ;
Determine the intervals on which $f'$ is monotonic by examining the sign of $f''$ ;
Sketch the graph of $f$ .

We take the derivative,
$f'(x) = 2 \sin x \cos x = \sin (2x).$

Thus,

$f'(x) = 0 \quad \implies \quad \sin (2x) = 0 \quad \implies \quad x = \frac{n \pi}{2}$

for $n$ an integer. (We found the zeros of sine in this exercise, Apostol Section 2.8 Exercise #1.)
$f$ is increasing if $n \pi < x < \left( n + \frac{1}{2} \right) \pi$ and decreasing if $\left( n - \frac{1}{2} \right) \pi < x < n \pi$ .
Taking the second derivative,
$f''(x) = -2 \cos (2x).$

Thus, $f'$ is increasing if $\left( n - \frac{1}{4} \right) \pi < x < \left( n + \frac{1}{4} \right) \pi$ and decreasing if $\left( n + \frac{1}{4} \right) \pi < x < \left( n + \frac{3}{4} \right) \pi$ .
We sketch the curve,