CONCAVITY AND THE SECOND DERIVATIVE TEST

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CONCAVITY AND THE SECOND DERIVATIVE TEST

• Section 3.4

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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

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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

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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

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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.

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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 .

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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 ___________.

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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.

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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 1600s was
Fermat. Along with Pascal, he founded the
subject of

• Probability
• Number Theory
• Abstract Algebra
• All of the above.

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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

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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.

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The following function has a relative minimum at

• 6856.0
• 0.0