Chapter 14, Problem 32RE

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Cost** Suppose a company has two separate plants that manufacture the same item. Suppose

To determine

**To calculate:** The production allocation that minimizes the company’s total cost. Suppose a company has two separate plants that manufacture the same item. Suppose that *x* is the amount produced at plant *y* is the amount produced at plant

Explanation

**Given Information:**

The total cost function for the two plants

**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 total cost function for the two plants

The provided function is

Recall that, 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

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