# Thread: Use Lagrange multipliers to find the maximum and minimum

1. ## Use Lagrange multipliers to find the maximum and minimum

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. (If a value does not exist, enter NONE.)
f(x,y,z) = 3x – y – 3z;
x + y – z = 0,
x^2 + 2z^2 = 1

2. Originally Posted by sammygirl2791
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. (If a value does not exist, enter NONE.)
f(x,y,z) = 3x – y – 3z;
x + y – z = 0,
x^2 + 2z^2 - 1

find $\displaystyle \nabla h = 0$

where $\displaystyle h(x,y,z,\lambda_1 , \lambda_2) = 3x - y - 3z +\lambda_1 (x + y - z)+\lambda_2 (x^2 + 2z^2 - 1)$

3. I can' figure out how to find y

4. Originally Posted by sammygirl2791
I can' figure out how to find y
Show me your workings thus far.

5. Originally Posted by sammygirl2791
I can' figure out how to find y
It is true that y does not appear in the gradient. However, after you have found, say, z as a function of x, you still have the two constraints, x+ y- z= 0 and [tex]x^2+ 2z^2= 1.

The second allows you to solve for specific values of x and z. Put those into x+ y- z= 0 to find y.