Chapter 14.4, Problem 23E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Production** Suppose that

*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 *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

**Formula used:**

To calculate relative maxima and minima of the

(1) Find the partial derivatives

(2) Find the critical points, that is, the point(s) that satisfy

(3) Then find all the second partial derivatives and evaluate the value of *D* at each critical point, where

(a) If

(b) If

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

For a function

(1) When both derivatives are taken with respect to *x* is

(2) When both derivatives are taken with respect to *y* is

(3) When first derivative is taken with respect to *x* and second derivative is taken with respect to *y* is

(4) When first derivative is taken with respect to *y* and second derivative is taken with respect to *x* is

Power of

Chain rule for function

Constant function rule for a constant

Coefficient rule for a constant

**Calculation:**

Consider the problem, the production function for a product with *x* units of one input and *y* units of a second input is

The provided function is

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

Thus,

Thus,

And,

Thus,

Thus, the critical points are

Recall that, for a function *x* is *y* is *x* and second derivative is taken with respect to *y* is

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

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