Rolles Theorem

1
Chapter 3

• Rolles Theorem

2
Rolles Theorem

• If
• A function is continuous on a closed interval
  a,b, and
• Has a derivative on the open interval (a,b), and
• Has the same y-value at the endpoints, a and b
• Then
• There must be at least one value of x, call it c,
  between a and b where the function has a
  horizontal tangent.

3
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4
Find the value of c guaranteed by Rolles Thm for
f(x)1-x2 in the interval -1,1.

• Is f(x) continuous on -1,1?
• Does f(x) have a derivative on (-1,1)?
• Is f(-1) f(1)?

• Yes
• Yes
• Yes

5
Does f(x)sin x satisfy the three conditions of
Rolles Thm on the interval -p/2,3p/2?

• Is f(x) continuous on the interval?
• Does f(x) have a derivative on the interval?
• Is f(- p/2) f(3p/2)?

• Yes
• Yes
• Yes

6
Find the two x-intercepts of f(x) and show that
f(x) 0 at some point between the two intercepts
7
How many values of c are guaranteed by Rolles
Thm on the interval -10,10 for the function

• 4
• 5
• 6
• 7
• The Theorem does not apply

Answer e