Difference between revisions of "Math 22 Concavity and the Second-Derivative Test"
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|Then we have: <math>f''(-1)=24>0</math>, <math>f''(\frac{1}{2})=-3<0</math> and <math>f''(2)=24>0</math> | |Then we have: <math>f''(-1)=24>0</math>, <math>f''(\frac{1}{2})=-3<0</math> and <math>f''(2)=24>0</math> | ||
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− | |'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>( | + | |'''Step 4''': By the concavity test, <math>f(x)</math> is concave up in <math>(-\infty,0)\cup (1,\infty)</math> and <math>f(x)</math> is concave down in <math>(0,1)</math> |
|} | |} | ||
Revision as of 08:04, 31 July 2020
Formal Definition of Concavity
Let be differentiable on an open interval . The graph of is 1. Concave upward on when is increasing on the interval. 2. Concave downward on when is decreasing on the interval.
Test for Concavity
Let be a function whose second derivative exists on an open interval 1. If for all in , then the graph of is concave upward on . 2. If for all in , then the graph of is concave downward on .
Guidelines for Applying the Concavity Test
1. Locate the -values at which or is undefined. 2. Use these -values to determine the test intervals. 3. Determine the sign of at an arbitrary number in each test intervals 4. Apply the concavity test
Exercises: Find the second derivative of and discuss the concavity of its graph.
1)
Solution: |
---|
Step 1: , so |
Step 2: So , so the test intervals are and |
Step 3: Choose for the interval , and for the interval . |
Then we have: and |
Step 4: By the concavity test, is concave up in and is concave down in |
2)
Solution: |
---|
Step 1: , so |
Step 2: So, and , so the test intervals are and |
Step 3: Choose for the interval , for the interval and for the interval . |
Then we have: , and |
Step 4: By the concavity test, is concave up in and is concave down in |
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