Mean Value Theorem – Differential Calculus for GATE, ESE, AE/JE

Mean Value Theorem (MVT) is one of the most important results in Differential Calculus. It connects continuity and differentiability and is frequently asked in GATE, ESE, AE/JE exams.

Types of Mean Value Theorems

• Rolle’s Theorem
• Lagrange Mean Value Theorem (LMVT)
• Cauchy Mean Value Theorem (CMVT)

1. Rolle’s Theorem

Conditions

• f(x) is continuous in [a, b]
• f(x) is differentiable in (a, b)
• f(a) = f(b)

Conclusion

There exists at least one c ∈ (a, b) such that:

f’(c) = 0

2. Lagrange Mean Value Theorem (LMVT)

Conditions

• f(x) is continuous in [a, b]
• f(x) is differentiable in (a, b)

Conclusion

There exists c ∈ (a, b) such that:

f’(c) = [f(b) − f(a)] / (b − a)

3. Cauchy Mean Value Theorem (CMVT)

Conditions

• f(x), g(x) are continuous in [a, b]
• f(x), g(x) are differentiable in (a, b)
• g’(x) ≠ 0

Conclusion

There exists c ∈ (a, b) such that:

[f’(c) / g’(c)] = [f(b) − f(a)] / [g(b) − g(a)]

Geometrical Interpretation

LMVT states that there exists a point where the tangent to the curve is parallel to the chord joining two points.

Solved Examples

Example 1 (Rolle’s Theorem)

f(x) = x² − 4x + 4 in [0, 4]

f(0) = 4, f(4) = 4 → condition satisfied

f’(x) = 2x − 4
2c − 4 = 0 → c = 2

Answer: c = 2

Example 2 (LMVT)

f(x) = x² in [1, 3]

f’(x) = 2x

[f(3) − f(1)] / (3 − 1) = (9 − 1)/2 = 4

2c = 4 → c = 2

Answer: c = 2

Example 3 (CMVT)

f(x) = x², g(x) = x in [1, 2]

f’(x) = 2x, g’(x) = 1

[f(2) − f(1)] / [g(2) − g(1)] = (4 − 1)/(2 − 1) = 3

2c / 1 = 3 → c = 3/2

Answer: c = 3/2

Example 4

If f is continuous and differentiable, then LMVT guarantees:

At least one point where slope equals average slope

Shortcut Tricks

• Always check conditions first
• Rolle’s Theorem → f(a) = f(b)
• LMVT → slope formula
• CMVT → ratio form

Questions

Q1. Rolle’s theorem gives?

Answer: f’(c) = 0

Q2. LMVT represents?

Answer: Average slope equals instantaneous slope

Exam Tips

• Check continuity & differentiability carefully
• Don’t skip conditions (common mistake)
• Memorize formulas
• Practice PYQs

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