Find the largest box (with faces parallel to the coordinate axes) that can be inscribed in$\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{4}}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{9}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{25}}\mathbf{=}\mathbf{1}\mathbf{.}$

The largest box (with faces parallel to the coordinate axes) that can be inscribed in$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1$

Two or more independent variables, an unknown function that depends on those variables and partial derivatives of the unknown function with respect to the independent variables make up a partial differential equation.

From the given information, it is a typical optimization problem with constraint, where the volume (V) of the box is to lie inside the ellipsoid given by the equation$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1$and need to find the proportions that will maximize this volume.

So, with the help of Lagrange multiplier method.

$$\begin{gathered} \frac{X^2}{4}+\frac{y^2}{9}+\frac{Z^2}{25}=1 \\ V=xyz \end{gathered}$$ …… (1)

Now, by the Lagrange multiplier method,$F=xyz+\lambda \left(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}\right)$.

Take partial derivative with respect to x , y , and z as follows:

$\frac{\partial F}{\partial x}=yz+\frac{\lambda x}{2}=0$ ………. (2)

$\frac{\partial F}{\partial y}=xz+\frac{2\lambda y}{9}=0$ ………. (3)

$\frac{\partial F}{\partial z}=xy+\frac{2\lambda z}{25}=0$ ……… (4)

Now with three partial derivative equations and the original constraints equation will be solve for unknown proportions.

Now, with the help of equation (2).

$\lambda =-\frac{2yz}{x}$

Substitute for $\lambda$in equation (3).

$$\begin{gathered} xz+\frac{2}{9}\times \frac{-2yz}{x}\times y=0 \\ {x}^{2}z-\frac{4}{9}{y}^{2}z=0 \\ z\left(x^2-\frac{4}{9}{y}^2\right)=0 \\ x=\pm \frac{2}{3}y \end{gathered}$$

Now, substitute in equation (4) for x.

$$\begin{gathered} \frac{2}{3}{y}^2+\frac{2}{25}\times \frac{-2yz}{{\displaystyle \frac{2}{3}}y}z=0 \\ \frac{2}{3}{y}^2-\frac{6}{25}{z}^2=0 \\ y=\pm \frac{3}{5}z\Rightarrow x=\pm \frac{2}{5}z \end{gathered}$$

Now substituting in equation (1) for x and (0,0) .

$$\begin{gathered} \frac{1}{4}{\left(\frac{2}{5}z\right)}^2+\frac{1}{9}{\left(\frac{3}{5}z\right)}^2+\frac{z^2}{25}=1 \\ z=\pm \frac{5}{\sqrt{3}},y=\pm \frac{3}{\sqrt{3}},x=\pm \frac{2}{\sqrt{3}} \end{gathered}$$

Thus, the solutions are$x=\pm \frac{2}{\sqrt{3}},y=\pm \frac{3}{\sqrt{3}},z=\pm \frac{5}{\sqrt{3}}$.