Question #1

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

P(x,y) =32x+6.7y0.004x²0.025y² dollars

wherexis the number of pounds of Kisses andyis 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 #2

Find the partial derivative of

f(x, y) = 3x³ 4xy + y²

with respect to x at the point

(1, 7, 24).

Step 1

We start by finding the partial derivative of f with respect to x, (denoted

f_x(x, y)).

Therefore, take the derivative with respect to x and treat the variable y as a constant. For example, the middle term

4xy is viewed as the constant (4y) times (x).

f_x(x, y) =------------

Question #3( Please help with step 2)

If

z=x⁴8x²+3x+8y³2y+5,

find

z

x

and

z

y

.

Step 1

Forz=f(x,y), we find the partial derivative with respect tox,

z

x

,

by treating the variableyas a constant and taking the derivative ofz=f(x,y) with respect tox. Since we treatyas a constant, the terms

8y³,

2y, and5will each have a derivative of0

0

Use this information to find

z

x

.

z

x

=

4x316x+3

Step 2 (Please help)

To find the partial derivative ofzwith respect toy, we treatxas a constant. Thus, the terms

x⁴,8x²,3x, and5

will each have a derivative of_________ . Use this information to find

z

y

.

z

y

=