Question 5 - 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 5 Verify Mean Value Theorem for the function 𝑓(𝑥) = 𝑥2 in the interval [2, 4]. 𝑓(𝑥) = 𝑥2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since 𝑓(𝑥) is polynomial . it is continuous ∴ 𝑓(𝑥) is continuous at (2, 4) Conditions of Mean value theorem 𝑓(𝑥) is continuous at (𝑎, 𝑏) 𝑓(𝑥) is differentiable at (𝑎 , 𝑏) If both conditions satisfied, then there exist some c in (𝑎 , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) − 𝑓(𝑎))/(𝑏 − 𝑎)Condition 2 Since 𝑓(𝑥) is a polynomial . it is Differentiable ∴ 𝑓(𝑥) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^′ (𝑐) = (𝑓(4) − 𝑓(2))/(4 − 2) Conditions of Mean value theorem 𝑓(𝑥) is continuous at (𝑎, 𝑏) 𝑓(𝑥) is differentiable at (𝑎 , 𝑏) If both conditions satisfied, then there exist some c in (𝑎 , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) − 𝑓(𝑎))/(𝑏 − 𝑎) Condition 2 Since 𝑓(𝑥) is a polynomial . it is Differentiable ∴ 𝑓(𝑥) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^′ (𝑐) = (𝑓(4) − 𝑓(2))/(4 − 2) 2𝑐= (4^2 − 2^2)/2 2𝑐 = 12/2 2𝑐 = 6 𝒄 = 𝟑 Hence c = 3 ∈(𝟐, 𝟒) Hence, Mean value 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