   Chapter 4.6, Problem 9E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. f ( x ) = 1 + 1 x + 8 x 2 + 1 x 3

To determine

To sketch: The graph of f that reveals all important aspects of the curve and estimate the intervals of increase, decrease, and intervals of concavity.

Explanation

Given:

The function is f(x)=1+1x+8x2+1x3

Calculation:

The function is,

f(x)=1+1x+8x2+1x3 (1)

Differentiate both sides of the Equation (1)

f'(x)=01x28×2x33x4=1x216x33x4 (2)

Differentiate both sides of the Equation (2)

f"(x)=2x3+16×3x4+3×4x5=2x3+48x4+12x5 (3)

Equate Equation (1) equal to 0.

f(x)=01+1x+8x2+1x3=0 (4)

Solve Equation (4).

x=0.126754

Equate Equation (2) equal to 0.

f(x)=01x216x33x4=0 (5)

Equate Equation (3) equal to 0.

f(x)=02x3+48x4+12x5=0 (6)

Draw the graph of the function f by substituting different values for x.

The sketches of the function f with the important aspects are shown in Figure 1and Figure 2.

Refer figure 1 and figure 2.

The curve f is increasing on (15.81,0.19)

The curve f is decreasing on (,15.81) , (0.19,0) and (0,) .

The local minimum value occurs at x=15.81 .

The local maximum value occurs at x=0.19 .

Therefore, the intervals of increase is on (15.81,0.19)_ .

Therefore, the intervals of decrease is on (,15.81)_ , (0.19,0)_ and (0,)_ .

Calculate the extreme values (local maxima and local minima) as shown below:

Solve Equation (5).

x=861and8+61=15

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