Summary: It is shown that the steepest-descent and Newton’s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to \(O(\varepsilon^{-2})\) to drive the norm of the gradient below \(\varepsilon\). This shows that the upper bound of \(O(\varepsilon^{-2})\) evaluations known for the steepest descent is tight and that Newton’s method may be as slow as the steepest-descent method in the worst case. The improved evaluation complexity bound of \(O(\varepsilon^{-3/2})\) evaluations known for cubically regularized Newton’s methods is also shown to be tight.