ROLLES THEOREM AND THE EXTREME VALUE THEOREM

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ROLLES THEOREM AND THE EXTREME VALUE THEOREM

• Section 3.2

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When you are done with your homework, you should
be able to

• Understand and use Rolles Theorem
• Understand and use the Mean Value Theorem

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ROLLES THEOREM AN ILLUSTRATION
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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 .

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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

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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

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Can Rolles Theorem be applied to the function
shown below on the interval ?

1. Yes
2. No

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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

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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

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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

1. The MVT can be applied. c 1.
2. The MVT can be applied. c -1 or 1.
3. The MVT cannot be applied since f is not
   continuous on the closed interval.
4. The MVT cannot be applied since f is not
   differentiable on the open interval.

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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.
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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.

1. True
2. False