Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 4

Take each partial derivative while holding the other coordinate fixed. Differentiating the polynomial term by term givesSince the function is a polynomial, both orders of the mixed partial derivative agree everywhere.

The multivariable chain rule follows the change of both coordinates along the parametric curve:Substituting the partial derivatives from part (a), with $x=1+s^2$ and $y=e^s$, yieldsThe first term comes from changing $x$, and the remaining terms from changing $y$.

At the origin the needed partial derivatives areThe Taylor series through total degree two is thereforewhich givesHere total degree is what matters: $x^2{y}^2$ is quartic, not quadratic. The exact omitted polynomial is $y^3+x^2{y}^2+x^2{y}^4$, which verifies the stated Big O notation remainder directly.

A critical point requires $f_x=f_y=0$. Since $y^4+y^2+1>0$ for every real $y$, the first partial derivative forces $x=0$. The second then reduces to $3y^2-3=0$, so the only critical points are $(0,1)$ and $(0,-1)$.

At these points the Hessian matrix is diagonal:The first is a positive-definite matrix, giving a strict local minimum; the second is an indefinite matrix, giving a saddle point. Their values are $f(0,1)=-1$ and $f(0,-1)=3$. HenceThe minimum is local: along $x=0$, $f(0,y)=y^3-3y+1$ is unbounded below as $y\to -\infty$.