Derivative Test

First Derivative Test §

First derivative test is when f’ changes from negative to positive at x = a or positive to negative (has a minimum or maximum)

If f has a local minimum at x=a on domain $(b,c)$,  $f^{\prime }\left(a\right)=$ 0 or undefined, and f’$\left(x\right)$changes from negative to positive.

Second Derivative Test §

If f’(a) = 0 and f”(a) > 0 then f has a Minimum

If f’(a) = 0 and f”(a) < 0 then f has a Maximum

If f has a local minimum at x=a on the domain $\left(b,c\right)$ $f^{\prime }\left(a\right)=$ 0 and $f^{\prime \prime }\left(a\right)$> 0.