Optimization problem for several variables calculus For $a,b,c > 0$, let 
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$
How can I find the axes parallel box of maximal volume inscribed in this ellipsoid ?

Suppose that the corner of the box in the first octant is $(x,y,z)$. Then the volume is $8xyz$. We want to maximize $x^2y^2z^2$ subject to the condition $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.
By the Arithmetic Mean Geometric Mean Inequality, we have 
$$\frac{1}{3}=\frac{1}{3}\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right) \ge \sqrt[3]\frac{x^2y^2z^2}{a^2b^2c^2},$$
with equality precisely if $\frac{x^2}{a^2}=\frac{y^2}{b^2}=\frac{z^2}{c^2}$.
Since the sum of these terms is $1$, each is $\frac{1}{3}$, and therefore for the maximum we have $x=\frac{a}{\sqrt{3}}$, $y=\frac{b}{\sqrt{3}}$, $z=\frac{c}{\sqrt{3}}$. 
Remark: Alternately, we can use tools from the calculus. Lagrange multipliers work smoothly. 
In addition to the constraint equation, we get by partial differentiation that for a maximum we need to have
$$8yz=\lambda\frac{2x}{a^2}, 8zz=\lambda\frac{2y}{b^2},\quad 8xy=\lambda\frac{2z}{c^2}.$$ 
From the first two equations, we get
$$\frac{8yz}{8zx}=\frac{(2x)(b^2)}{(2y)(a^2)},$$
or equivalently $\frac{x^2}{a^2}=\frac{y^2}{b^2}$. Similarly, $\frac{y^2}{b^2}=\frac{z^2}{c^2}$, and we are essentially finished. 
