= Solution
For a twice continuously differentiable function, a finite second-order <Taylor expansion> about an interior point is
$$
f(x+a,y+b)=f(x,y)+f_xa+f_yb+\frac12(Aa^2+2Bab+Cb^2)+o(a^2+b^2),
$$
where $A=f_{xx}$, $B=f_{xy}$, $C=f_{yy}$ at that point. A differentiable interior extremum requires $f_x=f_y=0$. For a strict minimum guaranteed by the quadratic term, the <Hessian matrix> must be positive definite. Completing the square gives $Aa^2+2Bab+Cb^2=A(a+Bb/A)^2+(AC-B^2)b^2/A$. Thus sufficient nondegenerate minimum conditions are
$$
\boxed{f_x=f_y=0,\qquad A>0,\qquad D=AC-B^2>0.}
$$
If $D=0$, the second-order test is inconclusive; for example $x^4+y^4$ has a strict minimum at zero despite a zero Hessian. The finite Taylor argument only needs $C^2$ regularity. The source's infinite Taylor representation additionally needs suitable analyticity: a smooth <flat function>, such as $e^{-1/(x^2+y^2)}$ extended by zero at the origin, has every Taylor coefficient zero there but is positive nearby.
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