1. CONCAVITY AND THE SECOND DERIVATIVE TEST
Section 3.4

2. When you are done with your homework, you should be able to…
Determine intervals on which a function is concave upward or concave downward
Find any points of inflection of the graph of a function
Apply the Second Derivative Test to find relative extrema of a function

3. Fermat Newton Pythagoras Pascal
I lived from I developed Calculus. I called Calculus “Fluents and Fluxions”. I discovered the law of gravity. I generalized the Binomial Theorem. Who am I?
Fermat
Newton
Pythagoras
Pascal

4. Definition of Concavity
Let f be differentiable on an open interval I. The graph of f is concave upward on I if is increasing on the interval and concave downward on I if is decreasing on the interval.
concave down concave up

5. Theorem: Test for Concavity
Let f be a function second derivative exists on an open interval I.
If for all x in I, then f is concave upward in I.
If for all x in I, then f is concave downward in I.

6. Definition: Point of Inflection
Let f be a function that is continuous on an open interval and let be a point in the interval. If the graph of I has a tangent line at this point , then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point

7. Theorem: Point of Inflection
If is a point of inflection of the graph of f, then either or does not exist at Hmmm….so this means that c is a ________ __________ of the ___________.

8. Guidelines for Determining Concavity on an Interval I and Finding Points of Inflection
Locate the critical numbers of Use these numbers to determine the test intervals.
Determine the sign of at one test value in each of the intervals.
Use the theorem regarding the test for concavity to determine whether is concave upward or concave downward on each interval.
Examine the results of the test for a change in concavity to determine if there are any inflection points.

9. Find the points of inflection and discuss the concavity of the graph of the function
The function is concave upwards over its entire domain. There is a point of inflection at (0 , 3)
The function is concave upwards on and concave downwards on
The function is concave upwards over its entire domain and there are no points of inflection.
None of the above

10. The most famous algebraist of the 1600’s was Fermat
The most famous algebraist of the 1600’s was Fermat. Along with Pascal, he founded the subject of …
Probability
Number Theory
Abstract Algebra
All of the above.

11. 5 cards are selected without replacement from a standard 52 card deck
5 cards are selected without replacement from a standard 52 card deck. Find the probability that all the cards are spades. This is called a straight flush in poker.
Both B and D

12. Theorem: Second Derivative Test
Let f be a function such that and the second derivative of f exists on an open interval containing c.
If then f has a relative minimum at
If then f has a relative maximum at
If the test fails. That is, f may have a relative maximum, a relative minimum, or neither. In such cases, you can use the First Derivative Test.

13. The following function has a relative minimum at
6856.0
0.0