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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

Sketch the graph of a function that satisfies all of the given conditions.

f′(0) =f′(4) = 0, f′(x) = 1 if x < −1,

f′(x) > 0 if 0 < x < 2,

f′(x) < 0 if − 1 < x < 0 or 2 < x < 4 or x > 4,

lim x 2 f ( x ) = , lim x 2 + f ( x ) =

f″(x) > 0 if − 1 < x < 2 or 2 < x < 4,

f″(x) < 0 if x > 4

To determine

To sketch: The graph of the function which satisfy the following conditions f(0)=f(4)=0 , f(x)=1 if x<1 , f(x)>0 if 0<x<2 , f(x)<0 if 1<x<0 or 2<x<4 or x>4 , limx2f(x)= and limx2+f(x)= , f(x)>0 if 1<x<2 or 2<x<4 , and f(x)<0 if x>4 .

Explanation

Definition used:

Increasing/Decreasing Test:

“(a) If f(x)>0 on an interval, then f is increasing on that interval.

(b) If f(x)<0 on an interval, then f is decreasing on that interval”.

Concavity Test:

“(a) If f(x)>0 for all x in I, then the graph of f is concave upward on I.

(b) If f(x)<0 for all x in I, then the graph of f is concave downward on I”.

Graph:

The condition f(0)=f(4)=0 indicates that there exists the horizontal tangents at x=0,4 .

The condition f(x)=1 if x<1 indicates that f is a line whose slope is 1 on the interval (,1) .

By the above definition, the condition f(x)>0 if 0<x<2 indicates that f is increasing on (0,2) and f(x)<0 if 1<x<0 or 2<x<4 or x>4 indicates that (1,0)(2,4)(4,)

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