given the function: $$z=f(x,y)=e^{-x^2-2y^2}$$
I'd like to find a point where if I were to place a ball, it would roll towards the direction $(2,1,a)$ .
Also, at which point could I place the ball so it wouldn't move at all?
I assume I need to calculate the gradient, and then check at which point the maximum slope vector is $(-2,-1,-a)$, i.e the opposite of where the ball would roll.
And for the second question I assume the gradient is $(0,0,0)$.
Am I correct? Thank you.
For your second question, you are correct that the only location with zero gradient is the origin. The gradient is
$$[-2x e^{(-x^2-2 y^2)}, -4y e^{(-x^2-2 y^2)}]$$
which is only zero at $$(x,y)=(0,0)$$