Does a stationary point of $f(x, y)$ need to have a zero slope in every direction?

Question

Does a stationary point of $f(x, y)$ need to have a zero slope in every direction?
If so, then why is it sufficient to show $ $ $ $ $f_x(a, b)=f_y(a, b)=0$ $ $ $ $ to determine that $(a, b)$ is a stationary point of $f(x,y)$?

With that approach, aren't we merely showing that $f(x,y)$ has a zero slope in the $x-$ and $y-$ directions at the point $(a, b)$, as opposed to in all directions at the point $(a, b)$?

Answer 1

The formula for the slope of a function $f$ in the direction represented by vector $v$ is $v \cdot \nabla f$. If at some point $\nabla f=0$ (that is, both $f'_x=0$ and $f'_y=0$), then the slope is $v \cdot 0 = 0$. Thus, if the slope is zero in the $x$ and $y$ directions, it is zero in any direction.

Answer 2

If $f$ is differentiable at $(a,b)$, then $D_{(a,b)}f\equiv0\iff\nabla f(a,b)=0$.

Answer 3

Okay, I understand it now. I was missing the actually definition of a slope, which is given as

$f_x(a,b)\cos{\alpha}+f_y(a,b)\sin{\alpha}$

where $\alpha$ is an angle determining the direction of the slope.

Hence, if $f_x(a,b)=f_y(a,b)=0$, then the above expression is also zero, and the slope is indeed zero.

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Can someone please confirm?