Chapter 14.4, Problem 24E

### 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 x units of one input and y units of a second input result in P   =   40 x +   50 y   −   x 2 −   y 2 −   x y units of a product. Determine the inputs x and y that will maximize P. What is the maximum production?

To determine

To calculate: The inputs x and y that will maximize P and the maximum production. Suppose that x units of one input and y units of second input result in P=40x+50yx2y2xy units of a product.

Explanation

Given Information:

The provided function is P=40x+50yâˆ’x2âˆ’y2âˆ’xy.

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.

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, x units of one input and y units of second input result in P=40x+50yâˆ’x2âˆ’y2âˆ’xy units of a product.

The provided function is P(x,y)=40x+50yâˆ’x2âˆ’y2âˆ’xy.

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

Thus,

âˆ‚Pâˆ‚x=040âˆ’2xâˆ’y=0âˆ’y=2xâˆ’40y=40âˆ’2x

And,

âˆ‚Pâˆ‚y=050âˆ’2yâˆ’x=0x+2y=50

Now, calculate value of x and y.

Substitute 40âˆ’2x for y in x+2y=50.

x+2(40âˆ’2x)=50x+80âˆ’4x=50âˆ’3x=âˆ’30x=10

Since, y=40âˆ’2x, thus, y=40âˆ’2(10)=40âˆ’20=20

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