# help Calc 3 problem. Find extrema using legrange multipliers with two? constraints.

• Jul 20th 2012, 06:01 PM
theflaktivist
help Calc 3 problem. Find extrema using legrange multipliers with two? constraints.
Not sure what I'm doing wrong here.....when I put into wolfram, it said the answer was (1,1,sqrt(2))

Find the extreme values of
f(x,y,z)=xyz

x^2+y^2+z^2=4

g1=x^2+y^2+z^2-4

x+y=2

g2=x+y-2

so I took the gradient of the function to get:

(yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2xλ+μ)j+(2zλ)k
• Jul 21st 2012, 06:53 AM
tom@ballooncalculus
Re: help Calc 3 problem. Find extrema using legrange multipliers with two? constraint
Quote:

Originally Posted by theflaktivist
(yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2xλ+μ)j+(2zλ)k

You mean,

(yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2yλ+μ)j+(2zλ)k

... but that gives you 3 equations in 5 unknowns, e.g...

yz = 2xλ+μ

... etc. And the g1 and g2 conditions are the other 2 (of the 5 equations in 5 unknowns).

Set $\displaystyle \nabla f = \lambda \nabla g$ where $\displaystyle g(x,y,z) = x^2 + y^2 + z^2$. Alternatively, you can solve it in one step using AM-GM but idk if your calculus professor would approve...