1. Multivariate Optimization Problems
University of Memphis Dept Math Sciences Calculus III
Homework Assignment # 4
Multivariate Optimization Problems
Find all critical points for the bivariate function f(x,y) = 2x3 – 2y3 + 3xy + 4 , and use the second derivative test to classify each critical point as either a local maximum, a local minimum, or a saddle point.
Repeat this exercise for g(x,y) = 3x – x3 – 2y2 +y4.
Bonus = Use your results for one or both of the above functions to sketch the countour map (level curves) in the xy-plane.
# 1. First and Second Derivative Tests.
Let D be the region in the first quadrant of the xy-plane bounded by the coordinate axes and the line x + y = 4. Consider the problem of finding the extreme values of f(x,y) = 3x + 2xy + 7y + 10 over D.
(b) Find the maximum and minimum values of f over D.
(a)
Find the critical point for f and give two reasons why this does not solve the stated problem. (One reason involves second derivatives, the other reason involves D). x D y
# 2. Absolute Extrema on Bounded Domains.
(a) Find the shortest distance from the point (2, 0, –3) to the plane x + y + z =1.
(b) Find the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
# 3. Distance Problems.
(a) Find the maximum and minimum values of F(x,y,z) = x + 2y + 3z subject to the constraint G(x,y,z) = x2 + y2 + z2 =1. Do this problem two different ways: (i) plug G into F (ii) use Lagrange multipliers
(b)
# 4. Optimization with Constraint.
Consider the problem of minimizing f(x,y) = 2x2 + 5y2 subject to y – 2x = 5.
Sketch the level curves f(x,y) = c for c = 5, c = 10, and c = 20. Also sketch the graph of the constraint. (Note: the easiest way to do all this graphing is to plot the x and y intercepts.)
Find the point (x,y) which solves the stated problem. (Note: the easiest way to do this is to use Lagrange multipliers.) Use the graph to check your answer, and also use the graph to explain why your solution gives the minimum value for f rather than the maximum value.
# 5. Soup Can Problem.
A cylindrical soup can is to be constructed so that it holds a volume of 500 cm3. The circular top and bottom of the can use material which costs five cents per cm3, while the side of the can is rolled from a sheet of material which costs three cents per cm3.
(a) Write the cost function C(x,y), where x represents the radius of the can and y is the height Use Lagrange multipliers to minimize C subject to the constraint V = px2y = 500.
(b) Find quantities in your solution which are in the same ratio (5 to 3) as the unit costs.