= Solution
Write $A=f_{xx}$, $B=f_{xy}$, $C=f_{yy}$ at the <stationary point>, and let $D=AC-B^2\ne0$. Under the usual second-order <Taylor expansion> hypothesis, the <second-derivative test> gives
$$
\boxed{\begin{array}{c|c}
D>0,\ A>0&\text{strict local minimum}\\
D>0,\ A<0&\text{strict local maximum}\\
D<0&\text{saddle point}
\end{array}}
$$
Indeed the quadratic term is $(A\xi^2+2B\xi\eta+C\eta^2)/2$. For $A\ne0$, completing its square gives $[A(\xi+B\eta/A)^2+(D/A)\eta^2]/2$. The two coefficients have the same positive sign, the same negative sign, or opposite signs in the respective cases. A negative $D$ also forces the two <Hessian> <eigenvalues> to have opposite signs, including the case $A=0$.
There is a qualification to the printed assumptions: existence of second <partial derivatives> only at the point does not ensure the needed quadratic approximation. Continuous second <partial derivatives> in a neighborhood are sufficient. The <existence of second partial derivatives does not justify the Hessian test>: for example, the function $g=x^2+y^2-6x^2y^2/(x^2+y^2)$ away from the origin, extended by $g(0,0)=0$, has <Hessian> $2I$ at the origin but $g(t,t)=-t^2$ and $g(t,0)=t^2$. Thus the classification above uses the standard additional regularity assumption, which the smooth function in part (b) satisfies.
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