Maximize and minimize

If $f(x,y) = x^2y + 2xy + 12y^2$ , then $\nabla f \; = \; \left( \begin{array}{c} 2(x+1)y \\ x^2 + 2x + 24y \end{array} \right)$ and so $\nabla f = 0$ at $(0,0)$ , $(-2,0)$ and $(-1,\tfrac{1}{24})$ . The values of $f$ at these points are $0$ , $0$ and $-\tfrac{1}{48}$ . We shall see that the maximum and minimum values of $f$ that are achieved on the boundary of the ellipse are more extreme than this, and so $M$ and $m$ will be achieved on the boundary of the ellipse.

Substituting $x \; = \; 3\cos\theta - 1 \hspace{1cm} y \; =\; \tfrac34\sin\theta$ into the target function, we see that $f$ becomes a cubic function of $\sin\theta$ on the boundary of the ellipse. Maximizing/minimizing this cubic, we obtain that $M=7.5$ and $m = \tfrac{1}{18} (45 - 11 \sqrt{33})$ , which makes $M + m \approx \boxed{6.5}$