Question

Find the maximum rate of change of f(x, y) =
`x^(2) - xe^(2y)` at the point (2, 0).

Answer 1

The maximum rate of change occurs in the direction of the gradient vector at the given point, with magnitude equal to the norm of the gradient at that point.

`f(x,y) = x^2 - xe^(2y) \implies \\abla f(x,y) = \left\langle 2x - e^(2y), -2xe^(2y)\right\rangle \\\\ \implies \\abla f(2,0) = \left\langle 3, -4 \right\rangle \\\\ \implies \left\|\\abla f(2,0)\right\| = √(3^2 + (-4)^2) = \boxed{5}`