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:

grad f=(yz)i+(xz)j+(xy)k

grad g1=(2x)i+(2y)j+(2z)k

grad g2=i+j

grad **f=λ**grad g1+**μ**grad g2

(yz)i+(xz)j+(xy)k=(2x**λ+μ**)i+(2x**λ+μ**)j+(2z**λ**)k

Re: help Calc 3 problem. Find extrema using legrange multipliers with two? constraint

Quote:

Originally Posted by

**theflaktivist** (yz)i+(xz)j+(xy)k=(2x**λ+μ**)i+(2x**λ+μ**)j+(2z**λ**)k

You mean,

(yz)i+(xz)j+(xy)k=(2x**λ+μ**)i+(2y**λ+μ**)j+(2z**λ**)k

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

yz = 2x**λ+μ**

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

Push ahead...

Re: help Calc 3 problem. Find extrema using legrange multipliers with two? constraint

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...