Chapter 14.4, Problem 23E

### Mathematical Applications for the ...

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

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+1.5y2âˆ’0.09x3âˆ’0.01y3 tons.

Formula used:

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

(1) Find the partial derivatives âˆ‚zâˆ‚x and âˆ‚zâˆ‚y.

(2) Find the critical points, that is, the point(s) that satisfy âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)âˆ’(zxy)2=âˆ‚2zâˆ‚x2â‹…âˆ‚2zâˆ‚y2âˆ’(âˆ‚2zâˆ‚xâˆ‚y)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=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

(2) When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Chain rule for function f(x)=u(v(x)) is fâ€²(x)=uâ€²(v(x))â‹…vâ€²(x).

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(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+1.5y2âˆ’0.09x3âˆ’0.01y3 tons.

The provided function is P(x,y)=3.78x2+1.5y2âˆ’0.09x3âˆ’0.01y3.

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

Thus,

âˆ‚Pâˆ‚x=03.78(2x)âˆ’0.09(3x2)=07.56xâˆ’0.27x2=0x(7.56âˆ’0.27x)=0

Thus, x=0 or x=28.

And,

âˆ‚Pâˆ‚y=01.5(2y)âˆ’0.01(3y2)=03yâˆ’0.03y2=0y(3âˆ’0.03y)=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=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x), when both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y), when first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

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

Pxx=âˆ‚2Pâˆ‚x2=âˆ‚âˆ‚x(âˆ‚Pâˆ‚x)=âˆ‚âˆ‚x(7

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### (exex2)1/2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### Sum of Square (SS), variance and standard deviation

Statistics for The Behavioral Sciences (MindTap Course List)

#### Evaluate the integral. 7. 11earctany1+y2dy

Single Variable Calculus: Early Transcendentals

#### Using for |x| < 1,

Study Guide for Stewart's Multivariable Calculus, 8th

#### The graph at the right is the direction field for: a) y = x y b) y = xy c) y = x + y d) y = xy

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th