   Chapter 14.4, Problem 23E 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

Production Suppose that P =   3.78 x 2 +   1.5 y 2 −   0.09 x 3 −   0.01 y 2 tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production. What is the maximum production?

To determine

To calculate: The values of x and y that will maximize the production and the maximum production. Suppose that P=3.78x2+1.5y20.09x30.01y3 tons is the production function for a product with x units of one input and y units of a second input.

Explanation

Given Information:

The production function for a product with x units of one input and y units of a second input is P=3.78x2+1.5y20.09x30.01y3 tons.

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 problem, the production function for a product with x units of one input and y units of a second input is P=3.78x2+1.5y20.09x30.01y3 tons.

The provided function is P(x,y)=3.78x2+1.5y20.09x30.01y3.

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

Thus,

Px=03.78(2x)0.09(3x2)=07.56x0.27x2=0x(7.560.27x)=0

Thus, x=0 or x=28.

And,

Py=01.5(2y)0.01(3y2)=03y0.03y2=0y(30.03y)=0

Thus, y=0 or y=100.

Thus, the critical points are (0,0), (0,100), (28,0), and (28,100).

Recall that, for a function z(x,y), the second partial derivative, when both derivatives are taken with respect to x is zxx=2zx2=x(zx), when both derivatives are taken with respect to y is zyy=2zy2=y(zy), when first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

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

Pxx=2Px2=x(Px)=x(7

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