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Baniyas International Private School
)2‫روضة‬+1‫رياض االطفال (روضة‬
‫ بنات‬+ ‫) بنين‬9-1( ‫المرحلة االساسية‬
‫بنات‬+ ‫) بنين‬12 - 10( ‫المرحلة الثانوية‬
Subject: Maths
Topic : Concavity & 2nd Derivative
Grade: 12
Academic Year 2015-2016 (2nd Trimester)
Student Name: ……………………………………………………………
Worksheet No: 2
Date:
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Concavity and the Second Derivative Test
Concavity:
Definition: The graph of a differentiable function y = f (x) is concave up on an interval where f ' (x) is
increasing and concave down on an interval where f ' (x) is decreasing.
Question: How do find where f ' (x) is increasing or decreasing?
Answer: The same way we did for f (x). Take the derivative of f ' (x) and see where it is positive
(increasing) and negative (decreasing).
If we take the derivative of the derivative we have found the second derivative. So now we have the
second derivative test for concavity:
The Second Derivative Test for Concavity
Let f be a twice differentiable function on an interval I.
1. If f ' ' (x) > 0 on I, the graph of f over I is concave up.
2. If f ' ' (x) < 0 on I, the graph of f over I is concave down.
Definition: An inflection point is a point of the graph where the function changes from concave up to
concave down. We can find inflection points with the second derivative:
Find where f ' ' (x) = 0 and where f ' ' (x) does not exist.
Ex 1: Find where f (x) = − x3 + 3x2 − 2 is concave up/down.
Step 1: Find possible inflection points: f ' ' (x) = 0 and f ' ' (x) does not exist.
We can also use the second derivative to find maximums and minimums:
The Second Derivative Test for Local Extrema
Let f be a continuous function on [a, b] and c be a critical point in [a,b].
1. If f' ' ′(x) < 0 , then f has a local maximum of f (c) at x = c.
2. If f ' ' (x) > 0 , then f has a local minimum of f (c) at x = c.
3. If f ' ' (x) = 0 , then the test is inconclusive and you must use the First Derivative Test for Local
Extrema.
Professional Learning Community
Mathematics Department
Ex 2: For the function f (x) = x4 + 2x3 − 2 find all the extrema using the second derivative test and
indicate the intervals where the graph is concave up/down.
Ex 3: For the function f (x) = x2/3 − 3 find all the extrema using the second derivative test and
indicate the intervals where the graph is concave up/down
Professional Learning Community
Mathematics Department