Sign of Partial Derivatives

Question

I'm struggling with part B.

I would know how to do it for the second partial derivative with respect to x or y, as the rate of change of the derivative for both of those cases is more intuitive.

However, this strategy sort of breaks down for this case in particular - another way of viewing the question is the partial derivative with respect to x of the partial derivative with respect to y. When I think of the partial derivative with respect to y, I just think of the curve along the y-axis. So when asked about the partial derivative of this curve with respect to x, I can't see how the answer could be anything but 0.

Any help will be greatly appreciated, thanks in advance.

Think of this as a rate of rate of change. Draw the y-derivative vector in your head, and then slide it forward along the x axis. Is it tilting forward or backward? — A. Thomas Yerger, Dec 21 '17 at 23:38

score 1 · Answer 1 · answered Dec 21 '17 at 23:43

1

Hint

Consider the function $(x,0)\mapsto\frac{\partial f}{\partial y}(x,0)$.

We see that $\frac{\partial f}{\partial y}(0,0)=0$.

What happens to the values of this function as $x$ increases from $0$?

answered Dec 21 '17 at 23:43

symplectomorphic

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Sign of Partial Derivatives

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