Question

Find the maximum rate of change of f(x, y) = title="x^{2} - xe^{2y}" alt="x^{2} - xe^{2y}" align="absmiddle" class="latex-formula"> 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.

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