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 $\nabla f(x)= \lambda \nabla g(x)$.)