Gradient of $f(x,y) = x^2 + y^2$ at $(0,0)$

Question

Gradient is supposed to point in the direction of steepest ascent, but when I try to find the gradient of $f(x,y) = x^2 + y^2$ I get $(2x, 2y)$ and at $(0,0)$ this is $(0,0)$, but what does this gradient vector mean? When looking at $f(x,y)$ on a graph, I can see that any direction you go the graph will increase, so shouldn’t there be some direction the gradient points in?

Answer 1

Of course not. The function $f$ has a minimum at $(0,0)$. So, you should expect that the gradient there is $(0,0)$.