UoH 2015 Question 25: Multivariable calculus

Let $z = f(x_1, x_2)$. Let $f_i$ be first order partial derivative of $f(.,.)$ w.r.t. $x_i$ and $f_{ij}$ be second order partial derivative of $f(.,.)$ w.r.t. $x_i$ and $x_j$. Suppose $f_1 = 0 = f_2$ at some point $(x_1, x_2) = (a,b)$ and further let $f_{11} \cdot f_{22} > f_{12}^2$ at $(a,b)$. Consider the statements: (I) The function attains its maximum if $f_{11} < 0$ at $(a,b)$. (II) The function attains its minimum if $f_{22} > 0$ at $(a,b)$.