Determining Point Coordinates From a Given Vector Field

Question

I seem to have gotten stuck on exactly what to do for this problem. Below is the question, plus my guess right under

I decided to take the partial of the first point (with respect to x) and then take the second partial with respect to y, but now I'm stuck. Is this question even asking for partials? Could I grab a hint about what to do in this situation?

Answer 1

Hint

The square of the wind speed is

$$\Vert F\Vert^2=\frac{x^2+y^2}{e^{2(x^2+y^2-1)^2}}$$

This is function of $r^2=x^2+y^2$.

Try to find the maximum of it in regards with $r$. This avoids using $x,y$ coordinates and partial derivatives.

Answer 2

I find that, upon differentiating wrt $r$, the critical points are $r=0$ and $r=\pm\sqrt{\dfrac12\pm\dfrac{\sqrt2}2}$.

But $r=0$ gives speed $0$.

So I suppose we take $r=\sqrt{\dfrac12 +\dfrac{\sqrt2}2}$ as the only possibility.