   Chapter 3.3, Problem 27E

Chapter
Section
Textbook Problem

# 20-27 Sketch the graph of a function that satisfies all of the given conditions. f ( 0 ) = f ′ ( 0 ) = f ′ ( 2 ) = f ′ ( 4 ) = f ′ ( 6 ) = 0 , f ′ ( x ) > 0  if  0 < x < 2  or  4 < x < 6 , f ′ ( x ) < 0  if  2 < x < 4  or  x > 6 , f ′ ′ ( x ) > 0  if 0 < x < 1  or  3 < x < 5 , f ′ ′ ( x ) < 0  if  1 < x < 3  or  x > 5 , f ( − x ) = f ( x )

To determine

To sketch:

The graph satisfying the given conditions

Explanation

1) Concept:

a) Increasing or Decreasing test:

i. If f'x>0 then f is increasing

ii. If f'x<0 then f is decreasing

b) Concavity test:

i. If f"(x)>0 then the graph of f is concave upward

ii. If f"(x)<0 then the graph of f is concave downward

2) Given:

f0=f'0=f'2=f'4=0,

f'x>0 if 0<x<2 or 4<x<6

f'x<0 if 2<x<4 or x>6

f"(x)>0 if 0<x<1 or 3<x<5

f"(x)<0 if 1<x<3 or x>5

f-x=f(x)<

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