Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).
9. f (x,y,z) = xyz; x^2 + 2y^2 + 3z^2 = 6
I keep going in circles with this question. I get the gradient of f(x) and g(x) and make them equal each other while the gradient of g(x) is multiplied by lambda (the Lagrange multiplier), and then I multiply every equation on the left and right by a variable so that each equation on the left is xyz. I then try to do some algebraic manipulation but since I can't cancel out the sufficient number of variables I can't solve the problem.
To get the gradients you have to take the partial derivatives with respect to x, y and z for both f(x, y, z) = xyz and g(x, y, z) = x^2 + 2y^2 + 3z^2 = 6.
Then you have to multiply the gradient of g(x, y, z) by lambda and make the gradient of f equal the gradient of g (when g is multiplied by lambda).
Shouldn't it be z?
And I got as far as you've outlined. Now I have to solve for a variable and get a numerical value for a variable so I can solve the remaining ones. How do I do this?
Help plz, thanks.