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CONCAVITY AND THE SECOND DERIVATIVE TEST. Section 3.4. 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
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CONCAVITY AND THE SECOND DERIVATIVE TEST Section 3.4
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
I lived from 1642-1715. 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
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 intervaland concave downward on I if is decreasing on the interval. concave downconcave up
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.
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 fif the concavity of f changes from upward to downward (or downward to upward) at the point .
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 ___________.
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.
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
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.
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
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.
The following function has a relative minimum at • 6856.0 • 0.0
