Thread: 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=λgrad g1+μgrad g2


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

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).

Push ahead...

Set where . Alternatively, you can solve it in one step using AM-GM but idk if your calculus professor would approve...