Chapter 14.4, Problem 30E

### 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 Let x be the number of work-hours required and let y be the amount of capital required to produce z units of a product. Show that the average production per work-hour, z/x, is maximized when ∂ z ∂ x = z x Use z = f ( x ,   y ) and assume that a maximum exists.

To determine

To prove: The average production per work-hour, z/x, is maximized when zx=zx. where x is the number of work-hours required and y is the amount of capital required to produce z units of a product.

Explanation

Given Information:

The average production per work-hour, is given by z/x, where z=f(x,y), and x is the number of work-hours required and y is the amount of capital required to produce z units of a product.

Formula used:

The partial differentiation of z=f(x,y) with respect to x by holding y constant as:

âˆ‚âˆ‚x[f(x,y)]=fx(x,y)

The partial differentiate f(x,y) with respect to y by holding x constant as:

âˆ‚âˆ‚y[f(x,y)]=fy(x,y)

Proof:

Consider the average production per work-hour function is P=zx.

Since, z=f(x,y), so,

P=zx=f(x,y)x.

Thus, the function is P=f(x,y)x.

Since, the maximum exists, thus, âˆ‚Pâˆ‚x=0

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