The Mean Value Theorem (MVT) states that if a function ( f ) is continuous on the closed interval ( [a, b] ) and differentiable on the open interval ( (a, b) ), then there exists at least one point ( c ) in ( (a, b) ) such that
[
f'(c) = \frac{f(b) – f(a)}{b – a}.
]
We can use the MVT to prove various inequalities. Below are some examples demonstrating the application of MVT in proving inequalities.
Example 1: Proving ( f(x) ) is monotonic
Statement: Prove that if ( f'(x) > 0 ) for all ( x ) in ( (a, b) ), then ( f(x) ) is strictly increasing on ( (a, b) ).
Proof:
- Let ( x_1, x_2 \in (a, b) ) such that ( x_1 < x_2 ).
- By the Mean Value Theorem, there exists a point ( c ) in ( (x_1, x_2) ) such that:
[
f'(c) = \frac{f(x_2) – f(x_1)}{x_2 – x_1}.
]
- Given that ( f'(x) > 0 ) for all ( x ) in ( (a, b) ), we have ( f'(c) > 0 ).
- Thus:
[
\frac{f(x_2) – f(x_1)}{x_2 – x_1} > 0.
]
- Since ( x_2 – x_1 > 0 ), it follows that ( f(x_2) – f(x_1) > 0 ), or equivalently ( f(x_2) > f(x_1) ).
- This proves that ( f(x) ) is strictly increasing on ( (a, b) ).
Example 2: Proving the Mean Value Theorem leads to the Cauchy-Schwarz Inequality
Statement: Prove ( |f(a)|
\leq \sqrt{\int_a^b |f(t)|^2 dt} ) using MVT.
Proof:
- Let ( f: [a, b] \rightarrow \mathbb{R} ) be a continuous function.
- By the Cauchy-Schwarz inequality in the integral form, we have for any continuous function ( g(t) ):
[
\left(\int_a^b f(t) g(t) dt\right)^2 \leq \left(\int_a^b f(t)^2 dt\right) \left(\int_a^b g(t)^2 dt\right).
]
- Choose ( g(t) = 1 ) for ( t \in [a, b] ):
- Then ( \int_a^b g(t)^2 dt = \int_a^b 1^2 dt = b – a ).
- Therefore, we have:
[
\left(\int_a^b f(t) dt\right)^2 \leq (b-a) \int_a^b f(t)^2 dt.
]
- By applying MVT, there exists ( c \in (a, b) ) such that:
[
f(c) = \frac{1}{b-a} \int_a^b f(t) dt.
]
- Replacing the integrals in the Cauchy-Schwarz inequality gives:
[
|f(c)|^2 \leq \frac{1}{b – a} \int_a^b f(t)^2 dt.
]
This provides a bound for the value of the function via its integral form over the domain.
Conclusion
In each of these examples, we see how the Mean Value Theorem provides a framework to relate the values of a function at individual points and its overall behavior over an interval. By carefully applying the theorem and basic properties of calculus (like continuity and differentiability), we can prove various inequalities and insights about functions.