Chapter 4 Theorems

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Chapter 4Theorems

• EXTREMELY important for the AP Exam

2
Rolles Theorem

• In the first quadrant, mark a and b on the x
  axis.
• Plot points at f(a) and f(b) such that
• f(a) f(b)
• Connect the two points however you would like but
  NOT in a horizontal line.
• What do you notice?

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Rolles Theorem
f(a)f(b)
a
b
What do you see between a and b on each of these
graphs????
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Rolles Theorem

• Suppose that y f(x) is continuous at every
  point of the closed interval a, b and
  differentiable at every point of its interior (a,
  b). If f(a) f(b) then there is at least one
  number c in (a, b) at which
• f(c) 0

5
Mean Value Theorem

• This is a more generalized version of Rolles
  Theorem.
• First stated by Joseph-Louis Lagrange

6
Mean Value Theorem

• Suppose that y f(x) is continuous at every
  point of the closed interval a, b and
  differentiable at every point of its interior (a,
  b).
• There is some point c in (a, b) at which

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Mean Value theorem
a
b
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Mean Value Theorem

• Suppose that y f(x) is continuous at every
  point of the closed interval a, b and
  differentiable at every point of its interior (a,
  b).
• There is some point c in (a, b) at which

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Mean Value theorem
c
a
b
10
Extreme Value Theorem

• If f is continuous on a closed interval a, b
• Then f attains both an absolute maximum M
• And an absolute minimum m
• In a,b

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Extreme Value Theorem (cont)

• That is, there are numbers x1 and x2 in a, b
• With f(x1) m and f(x2) M
• And m f(x) M for every other x in a, b

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y f(x)
M
m
b
a
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y f(x)
M
m
a
b
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y f(x)
M
m
a
b
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