Use the level curves of the function to determine if each partial derivative at the point P is positive, negative, or zero.

Question

It is asking me to find $f_{xx}$, $f_{yy}$, $f_x$, $f_y$, but I'm really unsure of how to determine the characteristics of the partial derivative. Please give me some guidance, thanks.

Answer 1

Let $P = (a, b)$. Define $g(u,v) = f(a + u, b + v)$.

Then, we may make the following approximations:

1. $f_x (a, b) = g_x(0, 0) \approx g(1, 0) - g(0, 0)$
2. $f_{xx} (a, b) = g_{xx}(0, 0) \approx g_x(1, 0) - g_x(0, 0)$

You can use method (1) to compute the $g_x$ terms:

$g_{xx}(0, 0)\approx (g(2,0) - g(1, 0)) - (g(1, 0) - g(0, 0))$

You can use any $dx$ you like, but I chose $dx = 1$ for simplicity.