Fermat's theorem on stationary points ($f\text{ differentiable at an interior local extremum }c⟹f^{\prime }(c)=0$)

If a function is differentiable at an interior local minimum or local maximum, its derivative there is zero. At a local minimum $c$, for small $h>0$ the difference quotient $(f(c+h)-f(c))/h$ is nonnegative, while for small $h<0$ it is nonpositive. The one-sided limit of a function from each side equals $f^{\prime }(c)$, forcing $f^{\prime }(c)\ge 0$ and $f^{\prime }(c)\le 0$. Apply the same argument to $-f$ at a local maximum.

The converse fails: $f(x)=x^3$ has $f^{\prime }(0)=0$ but no local minimum or local maximum at zero. The interior hypothesis is essential; a minimum attained only at an endpoint need not have derivative zero.