Applications of Differentiation – Concavity

Applications of Differentiation – Concavity

Learning Objectives:

Understand Applications of Differentiation – Concavity

Applications of Differentiation – Concavity

A Point of Inflection is point on the function where the Concavity changes from Concave Up to Concave Down or Concave Down to Concave Up.

If (c, f(c)) is a point of inflection, then either f”(c) = 0 or f”(c) does not exist.

Procedure for determining Concavity:

1.     Determine the values for which f”(x) = 0 or is undefined.

2.     Use the values to determine the test intervals.

3.     Determine the sign of  f”(x).

4.     If f”(x) > 0, the interval is Concave Up. If f”(x) < 0, the interval is Concave Down.

5.     If intervals change sign, there is a point of inflection.

The 2nd Derivative test to determine relative extrema:

1.     Determine the critical values of the function by determining where f’(x) = 0 or does not exist.

2.     Find the corresponding y values of the critical values.

3.     Find f”(x) and determine the sign of f”(x) at the critical values.

4.     If f”(x) > 0, there is a relative minimum at the critical value. If f”(x) < 0, there is a relative maximum at the critical value. If f”(x) = 0, the test fails.

 

 

 

 

Hook question:

1.     What is Point of Inflection?

2.     State the procedure to determine Concavity.

3.     Discuss the 2nd Derivative test to determine Relative Extrema.

 

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