Use derivatives to sketch the graph of a function

by RoRi

September 29, 2015

Let

$f(x) = x^3 - 6x^2 + 9x + 5.$

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,
$f'(x) = 3x^2 - 12x + 9.$

Thus,

$f'(x) = 0 \quad \implies \quad 3x^2 - 12x + 9 = 0 \quad \implies \quad x = \{ 1, 3 \}.$
$f$ is increasing for $x < 1$ and $x > 3$ , and decreasing for $1 < x < 3$ .
Taking the second derivative,
$f''(x) = 6x - 12.$

Thus, $f'$ is increasing if $x > 2$ , and decreasing if $x < 2$ .
We sketch the curve,