= Solution
The multivariable <chain rule> gives, at $(x,y)=(a+t\cos\gamma,b+t\sin\gamma)$,
$$
g'(t)=f_x\cos\gamma+f_y\sin\gamma,
$$
$$
g''(t)=f_{xx}\cos^2\gamma
+2f_{xy}\sin\gamma\cos\gamma
+f_{yy}\sin^2\gamma.
$$
Sufficient conditions for a strict local minimum at $t=0$ are $g'(0)=0$ and $g''(0)>0$.
At a <stationary point>, $g'(0)=0$ in every direction. Its second derivative is the quadratic form of the <Hessian matrix>. If
$$
f_{yy}>0,\qquad f_{xx}f_{yy}-f_{xy}^2>0,
$$
then completing the square gives
$$
f_{xx}u^2+2f_{xy}uv+f_{yy}v^2
=f_{yy}\left(v+\frac{f_{xy}}{f_{yy}}u\right)^2
+\frac{f_{xx}f_{yy}-f_{xy}^2}{f_{yy}}u^2>0
$$
for every nonzero direction $(u,v)$. The Hessian is therefore <positive-definite matrix>[positive definite], and the stationary point is a strict local minimum.
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