Difference between revisions of "Math 22 Concavity and the Second-Derivative Test"
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==Points of Inflection== | ==Points of Inflection== | ||
− | If the graph of a continuous function has a tangent line at a point where its concavity changes from upward to downward (or downward to upward), then the point is a point of inflection. | + | If the graph of a continuous function has a tangent line at a point |
+ | where its concavity changes from upward to downward (or downward to upward), | ||
+ | then the point is a point of inflection. | ||
+ | |||
+ | If <math>(c,f(c))</math> is a point of inflection of the graph of <math>f</math>, then either <math>f''(c)=0</math> or <math>f''(c)</math> is undefined. | ||
[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' |
Revision as of 07:07, 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 |
Points of Inflection
If the graph of a continuous function has a tangent line at a point where its concavity changes from upward to downward (or downward to upward), then the point is a point of inflection. If is a point of inflection of the graph of , then either or is undefined.
This page were made by Tri Phan