Maximization problem on an ellipsoid 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

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)
