   Chapter 10.1, Problem 41E ### 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. y = 7.5 x 4 − x 3 + 2

To determine

To calculate: The location of critical points and determines viewing window that shows all features of the graph f(x)=7.5x4x3+2 and sketch the complete graph.

Explanation

Given Information:

The provided function is f(x)=7.5x4x3+2.

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 equation f(x)=7.5x4x3+2,

Use the simple power rule to differentiate,

dydx=30x33x2=3x2(10x1)

Equate the above derivative to 0,

y=03x2(10x1)=0

From the property of zero multiplication,

3x2=0x=0

Or,

(10x1)=0x=110=0.1

Evaluate the values of x from the equation:

3x2(10x1)=0x=0,0.1

Hence, the value(s) of x are x=0,0.1

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

Substitute 0 for x in f(x)=7.5x4x3+2.

y=7.5(0)4(0)3+2=00+2=2

Hence, (0,2) is a critical point.

Substitute 0.1 for x in f(x)=7.5x4x3+2.

y=7.5(0.1)4(0.1)3+2=0

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