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Question 3 - Verify Rolles theorem - Chapter 5 Class 12 Continuity and Differentiability
Last updated at August 25, 2026 by Teachoo
Verify Rolles theorem
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
- Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
- Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
- Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
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Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
Question 3 If ๐ : [โ 5, 5] โ ๐ is a differentiable function and if ๐ โฒ(๐ฅ) does not vanish anywhere, then prove that ๐ (โ5) โ ๐ (5). ๐ : [โ 5, 5] โ ๐ is a differentiable โ We know that every differentiable function is continuous. Therefore f is continuous & differentiable both on (โ5, 5) By Mean Value Theorem There exist some c in (5, โ5) Such that ๐^โฒ (๐)=(๐(๐) โ ๐(๐))/(๐ โ ๐) Given that ๐^โฒ (๐ฅ) does not vanish any where โ ๐^โฒ (๐ฅ) โ 0 for any value of x Thus, ๐^โฒ (๐) โ 0 (๐(5) โ ๐(โ5))/(5 โ(โ5) ) โ 0 (๐(5) โ ๐(โ5))/(5 + 5) โ 0 ๐(5)โ ๐(โ5) โ 0 ร 10 ๐(5)โ ๐(โ5) โ 0 ๐(5) "โ " ๐(โ5) Hence proved.
