Question

Medium is the shortest distance from the surface to the origin

Answer 1

Minimize x2+y2+z2x2+y2+z2 given g(x,y,z)=3x2+y2−4xz=4g(x,y,z)=3x2+y2−4xz=4

Let f(x,y,z,λ)=x2+y2+z2+λ(3x2+y2−4xz−4)f(x,y,z,λ)=x2+y2+z2+λ(3x2+y2−4xz−4)

Now, using Lagrange Multiplier Method,

∂f∂x=2x+6λx−4λz=0∂f∂x=2x+6λx−4λz=0

∂f∂y=2y+2yλ=0∂f∂y=2y+2yλ=0

∂f∂z=2z−4xλ=0∂f∂z=2z−4xλ=0

Also 3x2+y2−4xz=43x2+y2−4xz=4

Solve these four equations in four variables, you will get the nearest point (x,y,z)(x,y,z)

But, check Hessian also to assure whether point gives minima or maxima or saddle point.