1. ## Help with Legrange

Problem:

Find max and/or min values of function f given the contraints:

f(x,y,z) = x^2 +y^2 +z^2

x + y + z = 1

x + 2y + 3z = 6

I know how to use legrange generally. When I solve using both constraints, I get values for x,y,z that when plugged into the function f gives 25/3.

The answer in the book gives: "No Maximum, minimum: 25/3"

My question is how do I know the value i got (25/3) is the minimum and how do I know there is no maximum for the function?

2. It sounds like you are doing the math work in the probelm right, which is good. One way to check your critical points is using the second partials of the function, which can be a pain, but in this case isnt too bad. Just remember to substitute a value for $\displaystyle z$ in terms of $\displaystyle x$ and $\displaystyle y$ in the equation before you try to take a derivative of it.

Equation: $\displaystyle D=FxxFyy-(Fxy)^2$
-If Fxx and D(x,y,z) are positve then its a min.
-If Fxx is negative and D(x,y,z) is positive then the point is a max.
-If D(x,y,z) is less than 0 then the point is a saddle point.

*note: where $\displaystyle Fxx$ means take the partial derivative of F with respect to x twice.

3. Yes but these equations have 3 variables so wouldnt I have to do something with the partial derivative with respect to z?

4. Correct, you would need to solve for Z in terms of x and y.
Then, you would put that in for F(x), then start the second partials.

After that your equatoin will be in terms of only x and y.