   Chapter 10.1, Problem 37E ### 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

# In Problems 37-42, use the derivative to locate critical points and determine a viewing window that shows all features of the graph. Use a graphing calculator to sketch a complete graph. f ( x ) = x 3 − 225 x 2 + 15 , 000 x 2 − 12 , 000

To determine

To calculate: The location of critical points and determines a viewing window that shows all features of the graph f(x)=x3225x2+15,000x12,000 also sketch the graph.

Explanation

Given Information:

The provided function is f(x)=x3225x2+15,000x12,000.

Formula Used:

To obtain the relative maxima and minima of a function,

1. Find the first derivative of the function.

2. Set the derivative equal to 0 to obtain the critical points.

3. Evaluate f(x) at values of x to the left and one to the right of each critical point to develop a sign diagram.

(a) If f(x)>0 to the left and f(x)<0 to the right of the critical value, the critical point is a relative maximum point.

(b) If f(x)<0 to the left and f(x)>0 to the right of the critical value, the critical point is a relative minimum point.

Calculation:

Consider the provided function f(x)=x3225x2+15,000x12,000,

Use the simple power rule to differentiate,

dydx=3x2450x+15000=3(x2150x+5000)=3(x100)(x50)

Equate the above derivative to 0,

y=dydx=3(x100)(x50)=0

Evaluate the values of x from the equation:

3(x100)(x50)=0x=100,50

Hence, the value(s) of x are x=100,50.

Evaluate the values of the original functions with the critical values:

Substitute 100 for x in the function f(x)=x315x216,800x+80,000.

y=(100)3225(100)2+15,000(100)12,000=10000002250000+150000012000=238,000

Hence, (100,238000) is a critical point.

Substitute 50 for x in the function f(x)=x315x216,800x+80,000.

y=(50)3225(50)2+15,000(50)12,000=125000225(2500)+15000(50)12000=300500

Hence, (50,300500) is a critical point

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