   Chapter 3.4, Problem 58E

Chapter
Section
Textbook Problem

# 57-60 Sketch the graph of a function that satisfies all of the given conditions. f ′ ( 2 ) = 0 ,     f ′ ( 0 ) = 1     f ′ ( x ) > 0  if  0 < x < 2 , f ′ ( x ) < 0  if  x > 2 ,     f ′ ′ ( x ) < 0  if  0 < x < 4 , f ″ ( x ) > 0  if  x > 4 ,     lim x → ∞ f ( x ) = 0 , f ( − x ) = − f ( x )  for all  x

To determine

To sketch:

The graph of function that satisfies all 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:

f'2=0, f'0=1,  f'x>0 if 0<x<2

f'x<0 if x>2,

f'x>0 if x>2,

f''x<0 if 0<x<4,

f''x>0 if x>4,

limxfx=0

f-x=-fx, for all x

3) Calculation:

The point 0, 0 in on the graph

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