Question 4 - Examples - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Question 4 Verify Rolle’s theorem for the function y = x2 + 2, a = – 2 and b = 2. y = x2 + 2, a = –2 and b = 2 Let 𝑓(𝑥) = 𝑥^2+2 Rolle’s theorem is satisfied if Condition 1 Since 𝑓(𝑥) is a polynomial, it is continuous ∴ 𝑓(𝑥) is continuous at (−2 , 2) Conditions of Rolle’s theorem 𝑓(𝑥) is continuous at (𝑎 , 𝑏) 𝑓(𝑥) is differentiable at (𝑎 , 𝑏) 𝑓(𝑎) = 𝑓(𝑏) If all 3 conditions satisfied then there exist some c in (𝑎 , 𝑏) such that 𝑓′(𝑐) = 0 Condition 2 Since 𝑓(𝑥) is a polynomial, it is differentiable ∴ 𝑓(𝑥) is differentiable at (−2 , 2) Also, 𝒇(−𝟐) = (−2)^2+2 = 4+2 = 6 𝒇(𝟐) = 2^2+2 = 4+2 = 6 Hence, 𝒇(𝟐) = 𝒇(−𝟐) Conditions of Rolle’s theorem 𝑓(𝑥) is continuous at (𝑎 , 𝑏) 𝑓(𝑥) is differentiable at (𝑎 , 𝑏) 𝑓(𝑎) = 𝑓(𝑏) If all 3 conditions satisfied then there exist some c in (𝑎 , 𝑏) such that 𝑓′(𝑐) = 0 Also, 𝑓(𝑥) = 𝑥^2+2 𝑓^′ (𝑥) = 2x So, 𝒇^′ (𝒄) = 𝟐𝒄 Since all 3 conditions are satisfied 𝒇^′ (𝒄) = 𝟎 2𝑐 = 0 𝒄 = 𝟎 Value of c i.e. 0 lies between −2 and 2. Hence c = 0 ∈ (−𝟐, 𝟐) Thus, Rolle’s theorem is satisfied.
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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo