= Existence of second partial derivatives does not justify the Hessian test
Define $f(0,0)=0$ and, elsewhere,
$$
f(x,y)=x^2+y^2-\frac{6x^2y^2}{x^2+y^2}.
$$
All first and second <partial derivatives> exist at the origin: $f_x=f_y=f_{xy}=f_{yx}=0$ and $f_{xx}=f_{yy}=2$. Thus its <Hessian matrix> there is the <positive-definite matrix> $2I$. Nevertheless $f(t,0)=t^2>0$ and $f(t,t)=-t^2<0$ for $t\ne0$, so the origin is a <saddle point of a scalar function>. A valid <second-derivative test> needs a second-order <Taylor expansion> with a remainder $o(x^2+y^2)$; continuous second <partial derivatives> in a neighborhood are a sufficient hypothesis. Existence of those derivatives at the point alone does not supply this hypothesis.
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