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Rolle's & LMVT — Mean Value Theorems

Applications of Derivatives

15

JEE Qs

8%

Hard

75

min

Always verify the continuity and differentiability conditions rigorously before attempting to apply Rolle's or LMVT.

🧮 Key Formulas

Rolle's Theorem: If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then exists c in (a,b) such that f'(c) = 0.
Lagrange's Mean Value Theorem (LMVT): If f(x) is continuous on [a,b] and differentiable on (a,b), then exists c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a).
Cauchy's Mean Value Theorem (CMVT): If f(x) and g(x) are continuous on [a,b] and differentiable on (a,b), and g'(x) != 0 for any x in (a,b), then exists c in (a,b) such that (f(b) - f(a)) / (g(b) - g(a)) = f'(c) / g'(c).

✅ Key Points for JEE

  • 1Always rigorously check all conditions (continuity and differentiability) before applying any MVT. Failure to satisfy even one condition makes the theorem inapplicable.
  • 2Rolle's Theorem is a special case of LMVT where the secant line between (a, f(a)) and (b, f(b)) is horizontal, implying f'(c)=0.
  • 3The 'c' guaranteed by the theorems is an *existence* result; finding its exact value usually involves solving an equation (f'(c)=0 or f'(c) = slope of secant).
  • 4Geometrically, LMVT states that there is at least one point 'c' where the tangent to the curve is parallel to the secant line joining the endpoints of the interval.
  • 5CMVT is a generalization of LMVT and is particularly useful in proving L'Hopital's rule and certain types of inequalities.

⚠️ Common Mistakes

  • ✕Failing to verify the continuity and differentiability conditions for the given function on the specified interval.
  • ✕Incorrectly calculating the derivative f'(x) or solving for the value of 'c' after applying the theorem.
  • ✕Assuming the theorem applies even when the function is not defined, not continuous, or not differentiable at certain points within the open interval (a,b) or at the endpoints.

📝 Practice Questions

See all

Q10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2

2025·MCQHard

Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________

2025·NumericalHard

Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3

2025·MCQHard

Q23.If the set of all values of a, for which the equation 5x3 −15x −a = 0 has three distinct real roots, is the interval (α, β), then β −2α is equal to ______

2025·NumericalMedium

Q13.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 1 cm/min. The surface area (in cm2 ) of the chocolate ball (without the ice- 4π cream layer) is : (1) 196π (2) 256π (3) 225π (4) 128π

2025·MCQMedium

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

2025·MCQHard

NCERT Chapters

  • Class 12 Maths Ch 5: Continuity and Differentiability
  • Class 12 Maths Ch 6: Applications of Derivatives

📚 Learn First

Derivatives of Standard Functions

Ability to calculate f'(x) is crucial for finding the point 'c' where the theorem holds.