   Chapter 3.3, Problem 28E

Chapter
Section
Textbook Problem

# Suppose f ( 3 ) = 2 , f ′ ( 3 ) = 1 2 , and f ′ ( x ) > 0 and f ′ ′ ( x ) < 0 for all x.(a) Sketch a possible graph for f.(b) How many solutions does the equation f ( x ) = 0 have? Why?(c) Is it possible that f ′ ( 2 ) = 1 3 ? Why?

To determine

(a)

To sketch:

The graph of f

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

2) Given:

f3=2

f'3=12

f'x>0

f"

To determine

(b)

To find:

How many solutions does the equation fx=0 have and why

To determine

(c)

To find:

Is it possible that f'2=13 and why

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