Assignment:
Partial Derivative – (Maximum & Minimum Values and Lagrange Multipliers)
Q1. Locate all relative maxima, relative minima, and saddle points of the surface defined by the following function.
f(x ,y) = 4x2y + 2y2 - 2xy2 -4y
Q2. Consider the minimization of f(x ,y) = x3 +y3 +3xy2 subject to the constraint of (x -3)2 + (y -3)2 = 0
(a) Draw the constraint curve on top of f(x , y) = (3, 3) with y-axis and x-axis between -2 and 6. Estimate where extrema values may occur and compute the function values corresponding to these extrema.
(b) Solve the problem in part (a) with the aid Lagrange multipliers. You may have to solve the equations numerically. Compare your answers with those in the part (a).
Q3. Let f(x, y, z) = 100 +x2 +y2 represent the temperature at each of the sphere x2 + y2 +z2 = 50 . Find the maximum temperature on the curve formed by the intersection of the sphere and the x + y + z = 5
Apply the method of Lagrange multipliers with two constraints. That is, maximize F(x, y , z, λ, μ ) = f(x , y, z) + λg (x , y, z) + μh (x , y, z) where g (x , y, z) = x2 +y2 +z2 -50 = 0 and where h (x , y, z) = x +y +z -5 =0
Q4. Find the absolute extrema of the following function on the indicated closed and bounded set R. f(x , y) = 5x2 +10y2 -5x x; R ={x2 + y2} ≤ 25
Q5. An international organization must decide how to spend the $2000 they have been allotted for famine relief in a remote area. They expect to divide the money between buying rice at $5/sack and beans at $10/sack. The number, P, people who would be fed if they buy x sacks of rice and y sacks of beans is given by P = x + 2y +5 * 10-8 x2 y2
Provide complete and step by step solution for the question and show calculations and use formulas.