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

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?

• Apparently, no. – Saucy O'Path Oct 28 '18 at 16:09

## 1 Answer

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

• I'm still a little bit confused, since I thought the gradient gives a direction to go for maximum increase, but the gradient isn't pointing anywhere in this example? – Ashton Halat Oct 28 '18 at 18:04
• Wouldn't the gradient at least be pointing in any one direction since it is a minimum though? Wouldn't a minimum mean that the gradient should have some direction for the function to increase? – Ashton Halat Oct 28 '18 at 23:56
• Which direction? It doesn't grow at that point. – José Carlos Santos Oct 28 '18 at 23:58