   Chapter 10.3, Problem 25E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# If the profit function for a product is P ( x ) = 5600 x + 85 x 2 − x 3 − 200 , 000 dollars, selling how many items, x, will produce a maximum profit? Find the maximum profit.

To determine

To calculate: The maximum profit of the provided profit function P(x)=5600x+85x2x3200,000 and the unit for which it is maximum.

Explanation

Given Information:

The provided profit function is:

P(x)=5600x+85x2x3200,000

Formula used:

If f(x) and g(x) are two differentiable functions then by the property of derivative:

ddx(f(x)+g(x))=ddxf(x)+ddxg(x)

And,

ddxxn=nxn1

Where, n is a constant and x is the variable.

To evaluate the relative maxima and minima of a function,

Step 1: Evaluate the critical values of the function.

Step 2: Now substitute the critical values into f(x) to find the critical points.

Step 3: Evaluate f(x) at each critical value for which f(x)=0.

If f(a)<0, a relative maximum occurs at a.

If f(a)>0, a relative minimum occurs at a.

If f(a)=0, or f(a) is undefined, the second derivative test fails, now use the first derivative test.

Calculation:

Consider the provided profit function:

P(x)=5600x+85x2x3200,000

The absolute maxima and absolute minima will occur only at the critical points.

To calculate the critical points of the provided function find the first derivative of the function:

P(x)=5600x+85x2x3200,000ddx(p(x))=ddx(5600x+85x2x3200,000)

If f(x) and g(x) are two differentiable functions then by the property of derivative:

ddx(

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