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Mathematical Applications for the ...

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

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BuyFindarrow_forward

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 4 240 x 3 + 16 , 200 x 2 60 , 000

To determine

To calculate: The location of critical points from the derivation of the function f(x)=x4240x3+16,200x260,000 and determine a viewing window that shows all features of the graph of f(x)=x4240x3+16,200x260,000 and sketch the complete graph.

Explanation

Given Information:

The provided function is f(x)=x4240x3+16,200x260,000.

Formula Used:

The simple power rule to derivative.

ddx(xn)=nxn1

The steps to calculate the critical values of a function

Step 1: First find the derivative of the function.

Step 2: Equate the derivate of the function to 0 and calculate the possible critical points.

Step 3: Put the critical point in the provided function to get the critical values.

Calculation:

Consider the provided function f(x)=x4240x3+16,200x260,000,

Use the simple power rule to differentiate,

dydx=4x3720x232400x=4x(x2180x+8100)=4x(x90)2

Equate the above derivative to 0,

y=04x(x90)(x90)=0

Evaluate the values of x from the equation:

4x(x90)2=0x=0,90

Hence, the value(s) of x are x=0,90.

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

Substitute 0 for x in f(x)=x4240x3+16,200x260,000.

y=(0)4240(0)3+16200(0)260,000=00+060,000=60,000

Hence, (0,60000) is a critical point.

Substitute 90 for x in f(x)=x4240x3+16,200x260,000.

y=(90)4240(90)3+16200(90)260,000=65610000174960000+13122000060000=21,810,000

Hence, (90,21,810,000) is a critical point

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