Lagrange multipliers

(a)

Centre the cuboid, with volume $V$ , with the lower face on the xy plane. Let the coordinates of the vertex in $x\ge0,y\ge0,z\ge 0$ where the cuboid intersects the hemisphere be $x, y, z$ .

$V=2x\,2y\,z$

The upper vertices are on the surface of the hemishpere so we get the constraint

$x^2+y^2+z^2=a^2 \quad(1)$

Form the objective function

$f=4xyz-\lambda \left(x^2+y^2+z^2-a^2 \right)$

Take partial derivates

$\begin{aligned} f_x &= 4yz - 2 \lambda x &= 0 \\ f_y &= 4xz - 2 \lambda y &= 0 \\ f_z &= 4xy - 2 \lambda z &= 0\\ f_{\lambda} &= -\left(x^2+y^2+z^2-a^2 \right) &= 0 \end{aligned}$