Thread: Max and min values of multivariable functions

In general I can figure out max and min values, but I'm have some trouble with the added constraints. Here's the problem:

"Find the maximum and minimum values of the function f(x,y)=x^2+y^2+z^2 subject to the constraint x^4+y^4+z^4=1"

Using the Lagrange multiplier method you can form an auxiliary function



The stationary points can then be found by solving

Do i include the partial derivative of gamma in the Gradient to find the points, or do i only use x, y, and z for the gradient still? Sorry we haven't covered much Lagrange yet. Just starting it actually.

Get them all. Why would we go to the trouble to define an extra variable and then just ignore it?

I think you mean lambda, yes take the derivative with respect to lambda and x,y,z.

You only need the result for x,y,z in context of the original function.

Yeah i meant lambda. I mix up greek letters quite a bit.

I didn't mean do I ignore it, I've just never seen a gradient include lambda before and I wasn't sure how to include it.

Once you start to studying the method it will become more clear why its included.

Good luck!

Okay so tell me if I get this correct or not. Pardon the lack of notation, I don't know how to type them.

(Gradient)L=<2x+4(lambda)x^3, 2y+4(lambda)y^3, 2z+4(lambda)z^3, x^4+y^4+z^4-1>

2x+4(lambda)x^3=0
2y+4(lambda)y^3=0
2z+4(lambda)z^3=0
x^4+y^4+z^4-1=0

x=sqrt(-1/(2*lambda))
y=sqrt(-1/(2*lambda))
z=sqrt(-1/(2*lambda))

(sqrt(-1/(2*lambda)))^4+(sqrt(-1/(2*lambda)))^4+(sqrt(-1/(2*lambda)))^4-1=0

lambda=sqrt(3)/2

And then i plug lambda back into the first three equations that are equal to 0.