For a twice-differentiable $f$, $f'(4) = 0$ and $f''(4) = 0$. What does the second derivative test say about $x = 4$?

Explanation

1. 01

   $f'' < 0$ at a critical point means a local maximum, $f'' > 0$ a local minimum.

2. 02

   When $f'' = 0$ the test decides nothing: use the first derivative test instead.