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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?

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    $\begingroup$ Apparently, no. $\endgroup$ – Saucy O'Path Oct 28 '18 at 16:09
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Of course not. The function $f$ has a minimum at $(0,0)$. So, you should expect that the gradient there is $(0,0)$.

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  • $\begingroup$ 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? $\endgroup$ – Ashton Halat Oct 28 '18 at 18:04
  • $\begingroup$ 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? $\endgroup$ – Ashton Halat Oct 28 '18 at 23:56
  • $\begingroup$ Which direction? It doesn't grow at that point. $\endgroup$ – José Carlos Santos Oct 28 '18 at 23:58

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