   Chapter 3.R, Problem 15E

Chapter
Section
Textbook Problem

# 13-15 Sketch the graph of a function that satisfies the given conditions. f  is odd ,      f ′ ( x ) < 0  for  0 < x < 2 , f ′ ( x ) >0 for  x > 2 ,     f ″ ( x ) > 0  for 0 < x < 3 , f ″ ( x ) <0 for  x > 3 ,     lim x →   ∞ f ( x ) = − 2

To determine

To sketch:

The graph of a function that satisfies the given conditions

Explanation

1) Concept:

i. Increasing or Decreasing test:

If f'x>0 then f is increasing

If f'x<0 then f is decreasing

ii. Concavity test:

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

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

iii. The function is odd when f-x= -f(x), which means that it is symmetric with respect to origin.

iv. The limit of the function as x tends to infinity is finite, then it stands for the horizontal asymptote of the function.

2) Given:

f is odd,f'x<0 for 0<x<2

f'x>0 for x>2,f''(x)>0 for 0<x<3

f''x<0 forx>3,

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