Use the slope to read the curve.

At x = a, the tangent has gradient $f'(a)$ and passes through (a, $f(a)$). If that gradient is non-zero, the normal gradient is −1/f′(a). A horizontal tangent has a vertical normal.

Stationary points have $f'(x)$ = 0. A change from positive to negative derivative gives a local maximum; negative to positive gives a local minimum. No sign change can give a stationary point of inflection. A global extremum on a closed interval also requires checking endpoints.