Question

Find the critical points and determine maxima or minima subject to the following constraint.
f(x,y,z) = 8x - 4z.
x^2+ 10y^2 + z^2 = 5.

Answer 1

Apply the Lagrange multiplier method. Then $L(x,y,z,\lambda)=8x-4z+\lambda(x^2+10y^2+z^2-5)$ then $L'_x=8+2\lambda x, L'_y=20\lambda y, L'_z=-4+2\lambda z$ . Solving the system $L'_x=0,L'_y=0,L'_z=0$

and $x^2+10y^2+z^2=5$ we get $x_1=-2,y_1=0,z_1=1, \lambda _1=2$ and $x_2=2,y_2=0,z_2=-1,\lambda_2=-2$ where $(-2,0,1),(2,0,-1)$ is critical points.

Findiing $L''_{x^2}=2\lambda,L''_{y^2}=20\lambda,L''_{z^2}=2\lambda,L''_{xy}=L''_{xz}=L''_{zy}=0$ we get

If $\lambda=2$ then $(-2,0,1)$ is point of minima constraint $f_{min}=-20$

If $\lambda=-2$ then $(2,0,-1)$ is point of maxima constraint $f_{max}=20$