Past exam of the mathematical methods II course of the University of Oxford / 2021 / 1

A stationary point occurs where the derivative vanishes. HereThe two families areConsequently there are infinitely many stationary points on the real line. The second derivative is $36\cos (3x)$: the first family consists of strict local minima and the second of strict local maxima.

The product rule and chain rule giveThe exponential function is strictly positive, so there are exactly two stationary points, at $\boxed{x=\pm 1/\sqrt{2}}$. The derivative is negative outside these two values and positive between them. Thus the negative value is a strict local minimum and the positive value a strict local maximum.

By the product rule and chain rule,Thus there are $\boxed{0}$ stationary points; the function is strictly increasing everywhere.