First we take the derivative. By some mildly tricky rewriting, we can factor this formula. Now that the derivative is nicely factored, we'll do the rest of the job. *f *' is undefined when *x* = 0, since
is undefined. Since *f *(0) *is* defined at 0, *x* = 0 is a critical point. Finally we want to find the roots of the derivative. We have *f *'(*x*) = *x*^{-1/3}(2 + 5*x*).
The factor *x*^{-1/3} is never 0. The factor (2 + 5*x*) is 0 when *x* = -5/2, so we've found another critical point. The final answer: there are critical points at *x* = 0 and at *x* = -5/2. If we zoom in enough on the graph, this makes sense. The derivative doesn't exist at 0, and there appears to be a horizontal tangent line at : |