Science:Math Exam Resources/Courses/MATH200/April 2012/Question 05 (a)
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Question 05 (a)
Let be the intersection of the plane
and the sphere
Use Lagrange multipliers to find the maximum value of on .
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Get the equations in this Lagrange multiplier problem by setting the partial derivatives of the objective function, , to zero, where
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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 .