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Question 3 - Verify Rolles theorem - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 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 𝑓 : [– 5, 5] → 𝐑 is a differentiable function and if 𝑓 ′(𝑥) does not vanish anywhere, then prove that 𝑓 (–5) ≠ 𝑓 (5). 𝑓 : [– 5, 5] → 𝐑 is a differentiable ⇒ We know that every differentiable function is continuous. Therefore f is continuous & differentiable both on (−5, 5) By Mean Value Theorem There exist some c in (5, −5) Such that 𝑓^′ (𝑐)=(𝑓(𝑏) − 𝑓(𝑎))/(𝑏 − 𝑎) Given that 𝑓^′ (𝑥) does not vanish any where ⇒ 𝑓^′ (𝑥) ≠ 0 for any value of x Thus, 𝑓^′ (𝑐) ≠ 0 (𝑓(5) − 𝑓(−5))/(5 −(−5) ) ≠ 0 (𝑓(5) − 𝑓(−5))/(5 + 5) ≠ 0 𝑓(5)− 𝑓(−5) ≠ 0 × 10 𝑓(5)− 𝑓(−5) ≠ 0 𝑓(5) "≠" 𝑓(−5) Hence proved.
