Use derivatives to sketch the graph of a function

Let

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

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,
$f(x) = x^2 - 3x + 2 \quad \implies \quad f'(x) = 2x - 3.$

Thus,

$f'(x) = 0 \quad \implies \quad 2x - 3 = 0 \quad \implies \quad x = \frac{3}{2}.$
Since $f'(x) < 0$ when $x < \frac{3}{2}$ and $f'(x) > 0$ when $x > \frac{3}{2}$ we have $f$ is decreasing if $x < \frac{3}{2}$ and $f$ is increasing if $x > \frac{3}{2}$ .
Taking the second derivative,
$f'(x) = 2x - 3 \quad \implies \quad f''(x) = 2.$

Since this is positive for all $x$ , this means $f'$ is increasing for all $x$ .
We sketch the curve,