Question

Find the maximum and minimum values of f(x,y,z)=3x+1y+5z on the sphere x^2+y^2+z^2=1

Answer 1

Use Lagrange multipliers to maximize g(x, y, z, ë) = 3x+1y+5z + ë(x^2+y^2+z^2-1) by differentiating it with respect to x, y, z and ë:

dg/dx = 3 + 2ëx = 0;
dg/dy = 1 + 2ëy = 0;
dg/dz = 5 + 2ëz = 0;
dg/dë = x^2+y^2+z^2-1 = 0

so
(a) 2ëx = -3
(b) 2ëy = -1
(c) 2ëz = -5
and (d) x^2+y^2+z^2 = 1

Divide (a) by (b) to get x/y = 3, or x = 3y
Divide (c) by (b) to get z/y = 5, or z = 5y

You can now express (d) entirely in terms of y^2: the rest should be plain sailing! Don't forget that square roots can take either positive or negative values.

Answer 2

Oops - that "ë" is supposed to be a lambda symbol. Wikipedia on "Lagrange Multipliers" will show how it works.