Maximize f(x,y,z) with Lagrange Multipliers

[Oglethorpe]

Find the maximum value of f(x,y,z) = 5xyz subject to the constraint of [PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP9619f6019f3fia87i60000567g3gb3dhi833if?MSPStoreType=image/gif&s=6&w=126&h=20.

I know to find the partial derivatives of the function and the constraint. Then, set up f(x)=λg(x) and so forth for each partial. Everything else is beyond me...

 

[Undoubtedly0]

Greetings! Correct, then use your constraint, g(x), to give you the fourth equation. In other words, your equations will be:
f_x = λ g_x,
f_y = λ g_y,
f_z = λ g_z, and
g(x).
With four equations and four unknowns, you should be able to solve.

 

[HallsofIvy]

By the way, since the value of \lambda itself is not part of the answer needed, you may find it simplest to start by eliminating \lambda by dividing one equation by another.

(I would not say that you set f(x)= \lambda g(x). Rather you set \nabla f(x)= \lambda \nabla g(x).)