Question #1

Find the minimum value of

z = 3x² + 5y² 2xy

subject to the constraint

x + y = 40.

z =

Question #2

The Kirk Kelly Kandy Company makes two kinds of candy, Kisses and Kreams. The profit, in dollars, for the company is given by

P(x, y) = 10x + 6.4y 0.001x² 0.025y²

where x is the number of pounds of Kisses sold per week and y is the number of pounds of Kreams. What is the company's profit if it sells the following amounts? (Round your answers to the nearest cent.)

(a) 80 pounds of Kisses and 6 pounds of Kreams $ ----------- (b) 120 pounds of Kisses and 24 pounds of Kreams $ ------------- (c) 0 pounds of Kisses and 256 pounds of Kreams $ --------------

Question #3

Suppose that the profit from the sale of Kisses and Kreams is given by

P(x, y) = 32x + 6.7y 0.004x² 0.025y² dollars

where x is the number of pounds of Kisses and y is the number of pounds of Kreams. Selling how many pounds of Kisses and Kreams will maximize profit? What is the maximum profit?

Step 1

The graph of the profit function is a three dimensional surface. For a maximum to exist, a horizontal plane will be tangent to the surface at that point, which means that the lines on the plane that are parallel to both horizontal axes will also be horizontal. Thus, both first partial derivatives must be equal to zero at any maximum value of the function. Begin by finding the first partial derivatives.

P_x=

P_y=

Question #4

If

z = x⁴ 8x² + 3x + 8y³ 2y + 5,

find

z

x

and

z

y

.

Step 1

For z = f(x, y), we find the partial derivative with respect to x,

z

x

,

by treating the variable y as a constant and taking the derivative of z = f(x, y) with respect to x. Since we treat y as a constant, the terms

8y³,

2y, and 5 will each have a derivative of . Use this information to find

z

x

.

z

x

=