   Chapter 9.6, Problem 31E ### 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 29 and 30, complete the following for each function.(a) Find f'(x).(b) Check your result in part (a) by graphing both it and the numerical derivative of the function.(c) Find x-values for which the slope of the tangent is 0.(d) Find points (x, y) where the slope of the tangent is 0.(e) Use a graphing utility to graph the function and locate the points found in part (d). f ( x ) = ( x 2 − 4 ) 3 + 12

(a)

To determine

To calculate: The derivative f(x) of the function f(x)=(x24)3+12.

Explanation

Given Information:

The function is f(x)=(x24)3+12.

Formula used:

According to the power rule, if f(x)=xn, then,

f(x)=nxn1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

g(x)=cf(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f(x)=u(x)+v(x)

According to the product rule, if f(x)=u(x)v(x), then

f(x)=u(x)v(x)+v(x)u(x)

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nun1dudx

Calculation:

Consider the provided function,

f(x)=(x24)3+12

Consider (x24) to be u,

f(x)=u3+12

Differentiate both sides with respect to x,

f(

(b)

To determine

To graph: The derivative f(x) of the function f(x)=(x24)3+12 and the numerical derivative of the function to check the results in part (a).

(c)

To determine

To calculate: The x-values for which the slope of the tangent of the function f(x)=(x24)3+12 is 0.

(d)

To determine

To calculate: The points (x,y) where the slope of the tangent of the function f(x)=(x24)3+12 is 0.

(e)

To determine

To graph: The function f(x)=(x24)3+12 and locate the points for which the slope of the tangent of the function is 0 using graphing utility.

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