Past exam of the mathematical methods II course of the University of Oxford / 2019 resit / 9 / a / ii

Differentiate to obtain $g^{\prime }(y)=-1-4\sin y$. Let $\alpha =\arcsin (1/4)$, with $0<\alpha <\pi /2$. All solutions of $g^{\prime }=0$ areSince $g^{\prime \prime }(y)=-4\cos y$, the first family consists of local maxima and the second of local minima. Their function values follow from $4\cos \alpha =\sqrt{15}$:The derivative is periodic, but $g$ itself is not: increasing $y$ by $2\pi$ lowers $g$ by $2\pi$, which accounts for the indexed vertical offsets.