   Chapter 3.3, Problem 74E

Chapter
Section
Textbook Problem

# Suppose that f′′′ is continuous and f ′ ( c ) = f ′ ′ ( c ) = 0 , but f ′ ′ ′ ( c ) > 0 . Does f have a local maximum or minimum at c? Does f have a point of inflection at c?

To determine

To explain:

The function f has or does not have a local maximum or minimum at c, also f has or does not have a point of inflection at c

Explanation

1) Concept:

i. 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

ii. The First Derivative Test-Suppose that c is a critical number of a continuous function f

If f' changes from positive to negative at c, then f has a local maximum at c

If f' changes from negative to positive at c, then f has a local minimum at c

If f' is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c

2) Definition:

A point P on a curve y=fx is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P

3) Given:

f''' Is continuous and f'c=f''c=0 ,f'''c>0

4) Calculation:

Given that f''' is continuous and f

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