Past exam of the mathematical methods II course of the University of Oxford / 2015 / 9 / a

The first partial derivatives areIf $\cos x=0$, then $z_x=-\sin x{e}^{-x^2-y^2}\ne 0$, so no stationary point is lost by excluding those values. The remaining condition $z_y=0$ forces $y=0$, and $z_x=0$ then givesThere are infinitely many such points. On every tangent branch $(n\pi -\pi /2,n\pi +\pi /2)$, the function $\tan x+2x$ has derivative ${\sec }^{2}x+2>0$ and goes from $-\infty$ to $+\infty$. The intermediate value theorem and strict monotonicity therefore give exactly one root on every branch. The central root is $x_s=0$, and oddness pairs the other roots at opposite signs.