This problem involves solving a multivariate optimization problem with constraints using the Lagrange method. Let’s break this down step by step.

Problem:

Maximize the function:
$f(x, y) = -2x^2 - 2y^2 + xy$ subject to the constraint:
$g(x, y) = y - 3x - 16 = 0$

Step 1: Define the Lagrange Function

The Lagrange function is:
$\mathcal{L}(x, y, \lambda) = f(x, y) + \lambda g(x, y)$ Substituting $f(x, y)$ and $g(x, y)$:
$\mathcal{L}(x, y, \lambda) = -2x^2 - 2y^2 + xy + \lambda (y - 3x - 16)$

Step 2: Compute the Partial Derivatives

We compute the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$, and set them to 0.

Partial derivative with respect to $x$: