   Chapter 14.4, Problem 20E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Profit The profit from the sales of two products is given by P ( x , y )   =   20 x +   70 y   −   x 2 −   y 2 dollars where x is the number of units of product 1 sold and y is the number of units of product 2. Selling how much of each product will maximize profit? What is the maximum profit?

To determine

To calculate: The number of units of each product sold to maximize the profit and the maximum profit. The profit from the sales of two products is given by P(x,y)=20x+70yx2y2 dollars, where x is the number of units of product 1 and y is the number of units of product 2.

Explanation

Given Information:

The provided function P(x,y)=20x+70yx2y2.

Formula used:

To calculate relative maxima and minima of the z=f(x,y),

(1) Find the partial derivatives zx and zy.

(2) Find the critical points, that is, the point(s) that satisfy zx=0 and zy=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)(zxy)2=2zx22zy2(2zxy)2.

(a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0.

(b) If D<0, then neither a relative maximum nor a relative minimum occurs.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by fx and with respect to y is denoted by fy.

For a function z(x,y), the second partial derivative,

(1) When both derivatives are taken with respect to x is zxx=2zx2=x(zx).

(2) When both derivatives are taken with respect to y is zyy=2zy2=y(zy).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy).

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Chain rule for function f(x)=u(v(x)) is f(x)=u(v(x))v(x).

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

Consider the function P(x,y)=20x+70yx2y2.

Use the power of x rule for derivatives, the constant function rule, the chain rule, and the coefficient rule,

Thus,

Px=0202x=02x=20x=10

And,

Py=0702y=02y=70y=35

Thus, the critical point is (10,35)

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