How can I interpret the second derivative test for this special case?

Question

Under certain conditions,it's known that the second derivative of a function of two variables in the direction of $\mathbf{u}=\langle h,k \rangle$ at a point P is $D_u\,^2f=f_{xx} \left( h+\frac{f_{xy}}{f_{xx}}k\right)^2+ \frac{k^2}{f_{xx}}\left(f_{xx}f_{yy} - f_{xy}^2\right)$ with $D=\left(f_{xx}f_{yy} - f_{xy}^2\right)$. It's said that if $D=0$, then would be uncertain to affirm that the function has any local or saddle point, but what if it was also given that $f_{xx}>0$, couldn't I deduce that we would necessarily have a local minima at P? Because $D_u\,^2f \geq 0$, being 0 only when $h+\frac{f_{xy}}{f_{xx}}k = 0$ (which I don't even know if this is even be possible) .