Question 05 (a)
Let be the intersection of the plane
and the sphere
Use Lagrange multipliers to find the maximum value of on .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Get the equations in this Lagrange multiplier problem by setting the partial derivatives of the objective function, , to zero, where
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We wish to maximize the value of subject to the constraints
So we use Lagrange multipliers. The objective function, , is
Computing the partial derivatives of with respect to all the variables and the Lagrange multipliers and setting them equal to zero gives:
To begin with, we observe that can not be zero, because if it was, then equations (1) and (2) would force , while equation (3) would force ; a contradiction. Hence, we combine equations (1) and (2) to see that the maximum of occurs on the plane :
Taking this result and using (4), we find
Taking these two results and using (5) we have an equation for z:
Multiplying this equation by 4 and expanding all the quadratic terms we get:
Solving this equation for we get
Taking the larger of the two results we obtain the maximum of subject to the given constraints which is .