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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

9–10 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:

1) produce graphs of f that reveal all the important aspects of the curve.

2) Estimate the intervals of increasing, decreasing, and concavity of a function

Explanation

1) Concept:

i) Function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

ii) If f''x>0 function is concave up and f''x<0 function is concave down in that particular interval.

2) Calculation:

We have,

fx=1+1x+8x2+1x3

The domain of the function is (-,)

Differentiate both sides with respect to x,

f'x=ddx1+1x+8x2+1x3

f'x=-1x2-16x3-3x4

Graph of f(x)

From graph observation

The interval of increasing is -, -0.20 and interval of decreasing are -0.20 , 0 and (0,)

The graph is concave downward in the interval -,-24and -0.25,0

The graph is concave upward in the interval  -24, -0.25 and (0, )

Now, by calculus, find the exact intervals of increasing, decreasing and concavity.

To find the exact value,

f'x=-1x2-16x3-3x4,   -1x2-16x3-3x4=0   x2+16x+3=0

x=-15.81, -0.19

Therefore, the intervals are -,-15.81, -1581,-0.19, -0.19,0,(0,)

for -,-15.81

 x=-20, f'-20=-1-202-16-203-3-204=-0.00048125<0

for -1581,-0.19

x=-1, f'-1=-1-12-16-13-3-14=12>0

for -0.19, 0x=-0.1, f'-0.1=-1-0.12-16-0

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