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Real Analysis

Notes 25.3 Second Mean Value Theorem

Let f and g be any two functions integrable in a,b and g be monotonic in a,b , then there exists

c  a,b such that
b c b
ò f(x) g(x) dx = g(a) f(x) dx g(b) f(x) dx
ò
+
ò
a a c
Proof: The proof is based on the following result known as Bonnet's Mean Value Theorem, given
by a French mathematician O. Bonnet [1819–1892].
Let f and g be integrable functions in [a,b]. If  is any monotonically decreasing function and
positive in [a,b], then there exists a point c  [a,b] such that
b c
ò f(x) (x) dx =  (a) g(x) dx.
ò

a a
Let g be monotonically decreasing so that  where (x) = g(x) – g(b), is non-negative and
monotonically decreasing in [a,b]. Then there exists a number c  [a,b] such that
b c
ò f(x)[g(x) g(b)]dx = [g(a) g(b)] f(x) dx
-
-
ò
a a
i.e.
b c b
ò
ò
ò f(x) g(x) dx = g(a) f(x) dx g(b) f(x) dx.
+
a a c
Now let g be monotonically increasing so that –g is monotonically decreasing. Then there exists
a number c  [a,b] such that
b c b
ò
-
-
ò f(x)[ g(x)]dx = - g(a) f(x) dx g(b) f(x) dx
ò
a
i.e.
b c h
ò f(x) g(x) dx = g(a) f(x) dx g(b) f(x) dx.
ò
ò
+
a a c
This completes the proof of the theorem.
There are several applications of the Second Mean Value Theorem. It is sometimes used to
develop the trigonometric functions and their inverses which you may find in higher
Mathematics. Here, we consider a few examples concerning the verification and application of
the two Mean Value Theorems.

x
Example: Verify the two Mean Value Theorems for the functions f(x) = x, g(x) = e in the
interval [–1, 1].
Solution: Verification of First Mean Value Theorem

Since f and g are continuous in [–1, 1], so they are integrable in [–1,1]. Also g(x) is positive in
[–1, 1]. By first Mean Value Theorem, there is a number  between the bounds of f such that

1 1 1 1
x
x
ò f(x) g(x) dx =  ò g(x) dx i.e., x e dx =  ò e dx.
ò
- 1 - 1 - 1 - 1
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