Maximization problem on an ellipsoid

Question

for three variables,

$$\max f(x,y,z)= xyz \\ \text{s.t.} \ \ (\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$

where $a,b,c$ are constant

how to solve the maximization optimization problem?

thank you for helpin

Answer 1

Hint: One way is symmetrization via change of variable $s=\frac{x}{a},t=\frac{y}{b},u=\frac{z}{c}$ and then optimize.

$\max f(s,t,u)=(abc)stu\,\,\text{subject to: } s^2+t^2+u^2=1.$ You can either use calculus or the fact that optimal point is of form $(\lambda,\lambda,\lambda)$. Thus $s=t=u=\frac{1}{\sqrt 3}$ so $(x,y,z)=(\frac{a}{\sqrt 3},\frac{b}{\sqrt 3},\frac{c}{\sqrt 3})$.

In order to verify your answer you can use Hessian matrix (second derivative test)