   Chapter 9.6, Problem 34E ### 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 31 and 32, do the following for each function f ( x ) .(a) Find f ' ( x ) .(b) Graph both f ( x ) and f ' ( x ) with a graphing utility.(c) Determine x-values where f ' ( x )   =   0 ,   f ' ( x )   >   0 ,   f ' ( x ) <   0 .(d) Determine x-values for which f ( x ) has a maximum or minimum point, where the graph is increasing, and where it is decreasing. f ( x ) = 3 + 1 16 ( x 2 − 4 x ) 4

(a)

To determine

To calculate: The derivative f(x) of the function f(x)=3+116(x24x)4.

Explanation

Given Information:

The function is f(x)=3+116(x24x)4.

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)=3+116(x24x)4

Consider (x24x) to be u,

f(x)=3</

(b)

To determine

To graph: The function f(x)=3+116(x24x)4 and its derivative f(x) with graphing utility.

(c)

To determine

To calculate: The x-values where f(x)=0,f(x)>0,f(x)<0 for the function f(x)=3+116(x24x)4.

(d)

To determine

To calculate: The x-values for which f(x)=3+116(x24x)4 has a maximum or minimum point, where the graph is increasing and where it is decreasing.

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