Chapter 14, Problem 8T

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Suppose a store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for $*x*, the type 2 sells for $*y*, and profit is given by

Find the selling prices that maximize profit. Find the maximum profit.

To determine

**To calculate:** The selling prices that maximize profit and the maximum profit where a store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for

Explanation

**Given Information:**

A store sells two brands of disposable razors and the profit for these is a function of their two selling prices. The type 1 razor sells for

**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 *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant and the partial derivative of *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant. The partial derivative of *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 type 1 razor sells for

The provided function is

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

Thus,

And,

Now, calculate the values of *x* and *y*

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