Use derivatives to sketch the graph of a function

by RoRi

September 29, 2015

Let

$f(x) = \frac{1}{x^2}.$

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,
$f'(x) = - \frac{2}{x^3}.$

Thus,

$f'(x) = 0 \quad \implies \quad -\frac{2}{x^3} = 0 \quad \implies \quad f'(x) \text{ is never zero}.$
$f$ is increasing for $x < 0$ , and decreasing for $x > 0$ .
Taking the second derivative,
$f'(x) = -\frac{2}{x^3} \quad \implies \quad f''(x) = \frac{6}{x^4}.$

Thus, $f'$ is increasing for all $x \neq 0$ (and is undefined at $x = 0$ ).
We sketch the curve,

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