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So I was doing this problem:

enter image description here

And the questions were what were the signs of $f_x,f_y,f_{xx},f_{yy},f_{xy}$. I thought I understood how to do the these types of questions but the answer for $f_{xx}$ was positive and $f_{yy}$ was also positive. I don't get why that is because isn't $f_{xx}$ decreasing at a decreasing rate. So shouldn't that make it negative?

I think I understand $f_{yy}$ since it's increasing at an increasing rate as we move up.

I just don't why the sign of $f_{xx}$ is positive.

Help me please!

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  • $\begingroup$ Consider a one-dimensional example $f(x)=x^2$, which has a positive second derivative. As you go from $x=-1$ to $x=1$, first it is decreasing at a decreasing rate, then it is increasing at an increasing rate. $\endgroup$
    – user856
    Dec 6 '17 at 5:09
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$f$ is decreasing at a decreasing rate as you move in the $x$ direction, which means $|f_x|$ is decreasing but $f_x$ is actually increasing (a negative number gets less negative by becoming more positive).

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