Math Tutoring on Applications of Differentiation – Concavity
A Point of Inflection is the 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 in Math Tutoring:
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.
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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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