Lagrange`s Mean Value Theorem

1. Statement of the Theorem

If a function [f(x)] satisfies:

1. It is continuous on the closed interval [[a,b]], and
2. It is differentiable on the open interval [(a,b)],

then there exists at least one point [c\in(a,b)] such that:

[f'(c)=\dfrac{f(b)-f(a)}{b-a}]

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2. Explanation (Why It Works)

• Since [f(x)] is continuous on [[a,b]], its graph has no breaks/jumps.
• Since [f(x)] is differentiable on [(a,b)], its graph is smooth (no sharp corners).
• There must be at least one point where the tangent is parallel to the secant joining [A(a,f(a))] and [B(b,f(b))].

This connects average rate of change and instantaneous rate of change.

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3. Geometrical Interpretation

• The line joining points [A(a,f(a))] and [B(b,f(b))] is the secant line with slope:

[\left(\dfrac{f(b)-f(a)}{b-a}\right)]

 

• LMVT ensures that at some point inside, the tangent line is parallel to this secant line.

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4. Important Points

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5. Important Formula

[f'(c)=\dfrac{f(b)-f(a)}{b-a}]

Where:

• LHS: Instantaneous rate of change
• RHS: Average rate of change

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6. Examples — Step-by-Step Solutions

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Example 1

Apply Lagrange`s Mean Value Theorem on [f(x)=x^{2}] on [[1,4]].

Solution:

1. [x^{2}] is continuous & differentiable everywhere → LMVT applies.
2. Compute RHS:

[\dfrac{f(4)-f(1)}{4-1}][=\dfrac{16-1}{3}][=\dfrac{15}{3}=5]

3. LHS: [f'(x)=2x]. Set:

[2c=5][ ⇒ c=\dfrac{5}{2}]

Conclusion: Lagrange`s Mean Value Theorem is satisfied at [c=\dfrac{5}{2}].

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Example 2

Apply Lagrange`s Mean Value Theorem to [f(x)=\sin x] on [[0,\pi]].

Solution:

1. Continuous & differentiable everywhere.
2. Compute RHS:

[\dfrac{f(\pi)-f(0)}{\pi-0}][=\dfrac{0-0}{\pi}=0]

3. Derivative: [\dfrac{d}{dx}(\sin x)=\cos x].
4. Solve: [\cos c=0][ ⇒ c=\dfrac{\pi}{2}]

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Example 3

Apply Lagrange`s Mean Value Theorem to [f(x)=e^{x}] on [[0,1]].

Solution:

1. Continuous & differentiable ⇒ okay.
2. Average slope:

[\dfrac{e^{1}-e^{0}}{1-0}][=e-1]

3. Derivative: [f'(x)=e^{x}].
4. Solve:

[e^{c}=e-1][ ⇒ c=\ln(e-1)]

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Example 4

Apply Lagrange`s Mean Value Theorem to [f(x)=\ln x] on [[1,e]].

Solution:

1. Continuous on [1,e], differentiable on [(1,e)].
2. RHS:

[\dfrac{\ln e-\ln 1}{e-1}][=\dfrac{1-0}{e-1}][=\dfrac{1}{e-1}]

3. Derivative: [\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}].

[\dfrac{1}{c}][=\dfrac{1}{e-1} ⇒ c=e-1]

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Example 5

Apply Lagrange`s Mean Value Theorem on [f(x)=x^{3}-3x+1] on [[-1,2]].

Solution:

1. Polynomial ⇒ continuous + differentiable.
2. Average slope:

[\dfrac{f(2)-f(-1)}{2+1}][=\dfrac{(8-6+1)-((-1)+3+1)}{3}][=\dfrac{3-(3)}{3}=0]

3. Derivative: [f'(x)=3x^{2}-3].
4. Solve:

[3c^{2}-3=0][ ⇒ c^{2}=1 ⇒ c=±1]

5. Select values inside (-1,2):

[c=1 \text{ (valid)},][\quad][ c=-1][ \text{ (endpoint, exclude)}]

Final Answer: [c=1].

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7. Conceptual Questions with Solutions

1. What are the conditions required for Lagrange`s Mean Value Theorem?

A function must be **continuous** on [[a,b]] and **differentiable** on [(a,b)].

2. What does Lagrange`s Mean Value Theorem guarantee?

There exists at least one real number [c\in(a,b)] such that [f'(c)=\dfrac{f(b)-f(a)}{b-a}].

3. What does the RHS term represent?

The **average rate of change** or slope of the secant line connecting [A(a,f(a))] and [B(b,f(b))].

4. What does the LHS term represent?

The **instantaneous rate of change** (slope of tangent line).

5. Why do we need continuity on [[a,b]]?

To ensure the graph has **no breaks or jumps**.

6. Why must the function be differentiable on [(a,b)]?

To ensure the graph is **smooth** with no sharp corners.

7. What happens if the differentiability condition fails?

LMVT **does not apply**.

8. Lagrange`s Mean Value Theorem is a generalization of which theorem?

**Rolle’s Theorem**. (When [f(a)=f(b)], LMVT reduces to Rolle’s theorem)

9. Can Lagrange`s Mean Value Theorem be applied to piecewise functions?

Yes — if continuity and differentiability conditions hold.

10. What if [f(b)-f(a) = 0]?

Then [\dfrac{f(b)-f(a)}{b-a}=0] and LMVT becomes Rolle’s theorem → [f'(c)=0].

11. Does Lagrange`s Mean Value Theorem guarantee only one such ‘c’ value?

No. It guarantees **at least one**. There may be more.

12. Can Lagrange`s Mean Value Theorem be applied to [|x|] on [-1,1]?

No, because [|x|] is **not differentiable** at [x=0].

13. Can Lagrange`s Mean Value Theorem be used on [\ln x] over [(-1,2)]?

No, because [\ln x] is **not defined / not continuous** on negative x.

14. What is the value of [c] for linear functions?

Any point in (a,b), because slope is **constant** throughout.

15. What does Lagrange`s Mean Value Theorem tell us about motion?

At some point, **instantaneous velocity = average velocity** over that interval.

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8. FAQ / Common Misconceptions

1. If function is continuous, Lagrange`s Mean Value Theorem always applies?

No. It must also be **differentiable** on [(a,b)].

2. A function with a sharp corner still satisfies Lagrange`s Mean Value Theorem ?

No. Sharp corner ⇒ not differentiable ⇒ LMVT fails.

3. Does Lagrange`s Mean Value Theorem find the exact maximum or minimum?

No. It finds where **slope of tangent equals slope of secant**.

4. Lagrange`s Mean Value Theorem can give values of c at endpoints?

No. [c] must lie **strictly between** [a] and [b].

5. Lagrange`s Mean Value Theorem only works for polynomials?

No. It works for **any** function satisfying the conditions.

6. Does Lagrange`s Mean Value Theorem apply to discontinuous derivatives?

Derivative must **exist** but need not be continuous everywhere.

7. If average slope is zero, does that mean flat function?

Not necessarily. Tangent is horizontal at **some point**, not everywhere.

8. Is Lagrange`s Mean Value Theorem same as Taylor theorem?

No. Taylor theorem is advanced and includes remainder terms.

9. Do we need function to be differentiable on endpoints?

No. Only inside interval [(a,b)].

10. Lagrange`s Mean Value Theorem is never used in real life?

Incorrect — used in **motion, economics, optimization**, etc.