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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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. In particular, you should use graphs of f′ and f″ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

f ( x ) = x x 3 + x 2 + 1

To determine

To sketch: The graph of f that reveals all important aspects of the curve and the graph of f' and f" to estimate the intervals of increase, decrease, the extreme values, the intervals of concavity and the inflection points.

Explanation

Given information:

The curve function is f(x)=xx3+x2+1 (1)

Calculation:

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

The sketch of the function f is shown in Figure 1.

Differentiate Equation (1) with respect to x.

f'(x)=(x3+x2+1)1x(3x2+2x)(x3+x2+1)2=x3+x2+13x32x2(x3+x2+1)2=2x3x2+1(x3+x2+1)2=(2x3+x21)(x3+x2+1)2 (2)

The sketch of the function f' is shown in Figure 2.

Differentiate Equation (2) with respect to x.

f"(x)=2x(3x4+3x3+x26x3)(x3+x2+1)3

The sketch of the function f" is shown in Figure 3.

Refer Figure 2.

The curve f is decreasing on (0.66,) .

The curve f is increasing on (,1.47) and (1.47,0.66) .

The local maximum value occurs at x=0.66 .

Therefore, the intervals of increase is on (,1.47)_ and (1.47,0.66)_ .

Therefore, the interval of decrease is on (0.66,)_

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