for critical values where f(x) is defined, the following tests are used (also called the first and second derivative tests for extrema) ...

first derivative test ...

f(x) has a max if f'(x) changes sign from (+) to (-) at that critical value.

f(x) has a min if f'(x) changes sign from (-) to (+) at that critical value.

second derivative test ...

if f''(x) > 0 at the critical value, then f(x) is concave down and a max exists at that critical value.

if f''(x) < 0, then f(x) is concave up and a min exists at that critical value.