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"
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
$\displaystyle L(x,y,z,\lambda) = (x^2+y^2+z^2)+\lambda (x^4+y^4+z^4-1)$
The stationary points can then be found by solving $\displaystyle \nabla L = 0$
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.
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.