Both Curvatures, Added

A scalar field is given by

$f(x,\,y) = x^{2}e^{3y} + 4\ln\!\left(x^{2} + 2y^{2}\right)$

For a function of two variables, the sum of the two pure second-order partial derivatives

$\nabla^{2} f = f_{xx} + f_{yy}$

measures how the field curves along the two axis directions at once. (The mixed partial $f_{xy}$ plays no part in it.)

Evaluate $\nabla^{2} f$ at the point $\left(1,\; \tfrac{1}{2}\right)$.

Use this value so that everyone works from the same constant:

Keep full precision through the intermediate steps and round only the final answer, to 2 decimal places.