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

For increments $h=x-x_s$ and $k=y-y_s$, the second-order multivariable Taylor expansion isAt a stationary point $(x_s,0)$, $\sin x_s=-2x_s\cos x_s$ and $z_s=\cos x_s{e}^{-x_s^2}$. Direct differentiation givesThus, since this particular function is analytic, the expansion can be writtenThere are no linear terms because the gradient vanishes, and no mixed quadratic term because the mixed derivative vanishes on the axis.