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ROLLES THEOREM AND THE EXTREME VALUE THEOREM

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When you are done with your homework, you should be able to... Understand and use Rolles Theorem ... If then there is at least one number c in such that . PROOF. Let ... – PowerPoint PPT presentation

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Title: ROLLES THEOREM AND THE EXTREME VALUE THEOREM


1
ROLLES THEOREM AND THE EXTREME VALUE THEOREM
  • Section 3.2

2
When you are done with your homework, you should
be able to
  • Understand and use Rolles Theorem
  • Understand and use the Mean Value Theorem

3
ROLLES THEOREM AN ILLUSTRATION
4
ROLLES THEOREM
  • Let f be continuous on a closed interval
  • and differentiable on the open interval
    . If then there is at least one
    number c in such that .

5
PROOF
  • Let
  • Case 1 If
    is a constant on the interval and
  • Case 2 Suppose
    . By the Extreme Value Theorem, you know that f
    has a maximum at some c in the interval. Since
    , this maximum does not occur at either
    endpoint. So f has a maximum in the open
    interval . This implies that is a
    relative maximum and thus a critical number of f.
    Finally, because f is differentiable at c, you
    can conclude that

6
PROOF CONTINUED
  • Case 3 Suppose
    . By the Extreme Value Theorem, you know that f
    has a minimum at some c in the interval. Since
    , this minimum does not occur at either
    endpoint. So f has a minimum in the open
    interval . This implies that is a
    relative minimum and thus a critical number of f.
    Finally, because f is differentiable at c, you
    can conclude that

7
Can Rolles Theorem be applied to the function
shown below on the interval ?
  • Yes
  • No

8
Show that Rolles Theorem be applied to the
function shown below on the given interval. What
is the exact value of c?
  • 2.5
  • 0.0

9
THE MEAN VALUE THEOREM
  • If f is continuous on a closed interval
  • and differentiable on the open
    interval then there exists at least one
    number c in the open interval such that

10
Determine whether the Mean Value Theorem can be
applied to f on the
closed interval If the MVT can be
applied, find all values of c in the open
interval such that
  • The MVT can be applied. c 1.
  • The MVT can be applied. c -1 or 1.
  • The MVT cannot be applied since f is not
    continuous on the closed interval.
  • The MVT cannot be applied since f is not
    differentiable on the open interval.

11
WORK IT OUT!!!a. Graph the function f on the
given interval.b. Find and graph the secant
line through points on the graph of f at
the endpoints of the given interval.c. Find
and graph any tangent lines to the graph of f
which are parallel to the secant line.
12
If the graph of a function has three
x-intercepts, then it must have at least two
points at which its tangent line is horizontal.
  • True
  • False
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