Verified answer

So you want to maximise x²+y²+z² subject to x+y+z=100.

The Lagrange function for this problem is:

Λ(x,y,z,λ) = x² + y² + z² + λ (x + y + z - 100)

Any optimal solutions must be stationary points of this function.

∂Λ/∂x = 0

∂Λ/∂y = 0

∂Λ/∂z = 0

∂Λ/∂λ = 0

Evaluating the partial derivatives, we find:

2x + λ = 0

2y + λ = 0

2z + λ = 0

x + y + z - 100 = 0

This is a system of linear equations, four equations in four unknowns. The unique solution is:

x = y = z = 100/3

λ = -200/3

So the only stationary point is if all three numbers are 100/3. Showing that this is a maximum is left as an exercise.