Quiz 7, Complete Solution for Problem 1

Because the values of x, y and z are non-negative, this is a little different from a standard Lagrange multiplier problem. The set of points satisfying the constraint is a plane; the set of points on the plane with non-negative coordinates forms a triangle. On the boundary of the triangle the function is zero, so the max must occur inside the triangle, since the function is positive at some of the points inside the triangle.

To find this maximum we can use the method of Lagrange multipliers. To do this, follow the procedure in Lagrange multipliers: A Review.

The first step is to find the gradient of the function we are trying to optimize and the gradient of the constraint function.

The gradient of f is:

The constraint is the relation between the variables x, y, and z. in this problem we know that x+2y+2z=108, so that is the constraint. Note that it is also in the form g(x,y,z)=c so we can use g(x,y,z)=x+2y+2z as the constraint function. The gradient of g is:

Remember that we want to find the points that satisfy the constraint equation, at which the gradient of f is parallel to the gradient of g (Why?). This gives the following system of equations:

So far we have one vector equation and one scaler equation.

When we re-write the first vector equation as a system of scaler equations we get the system of four equations:

One approach to solving these equations is to use the first equation to eliminate lambda from the other equations. When we do this we get:

If we divide the first equation by z and the second equation by y we get:

(Why can we divide by y and z without worrying about losing solutions? For the answer Click here )

So we get that 3x=108, and x=36, so y=18, z=18. This is the only critical point. Since f is zero when either x, y or z is zero, and f is positive here, f must have its maximum here. Maximum value of f=(18)(18)(36).

Return to Quiz 7.