Chapter 14.4, Problem 20E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Profit** The profit from the sales of two products is given by

*x* is the number of units of product 1 sold and *y* is the number of units of product 2. Selling how much of each product will maximize profit? What is the maximum profit?

To determine

**To calculate:** The number of units of each product sold to maximize the profit and the maximum profit. The profit from the sales of two products is given by *x* is the number of units of product 1 and *y* is the number of units of product 2.

Explanation

**Given Information:**

The provided function

**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 function

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

Thus,

And,

Thus, the critical point is

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