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Example 43 - Examples

Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 43 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Example 43 Verify Mean Value Theorem for the function 𝑓(π‘₯) = π‘₯2 in the interval [2, 4]. 𝑓(π‘₯) = π‘₯2 in interval [2, 4]. Checking conditions for Mean value Theorem Condition 1 Since 𝑓(π‘₯) is polynomial . it is continuous ∴ 𝑓(π‘₯) is continuous at (2, 4) Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž, 𝑏) 𝑓(π‘₯) is differentiable at (π‘Ž , 𝑏) If both conditions satisfied, then there exist some c in (π‘Ž , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž)Condition 2 Since 𝑓(π‘₯) is a polynomial . it is Differentiable ∴ 𝑓(π‘₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^β€² (𝑐) = (𝑓(4) βˆ’ 𝑓(2))/(4 βˆ’ 2) Conditions of Mean value theorem 𝑓(π‘₯) is continuous at (π‘Ž, 𝑏) 𝑓(π‘₯) is differentiable at (π‘Ž , 𝑏) If both conditions satisfied, then there exist some c in (π‘Ž , 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž) Condition 2 Since 𝑓(π‘₯) is a polynomial . it is Differentiable ∴ 𝑓(π‘₯) is differentiable in (2, 4) Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (2, 4) such that, 𝑓^β€² (𝑐) = (𝑓(4) βˆ’ 𝑓(2))/(4 βˆ’ 2) 2𝑐= (4^2 βˆ’ 2^2)/2 2𝑐 = 12/2 2𝑐 = 6 𝒄 = πŸ‘ Hence c = 3 ∈(𝟐, πŸ’) 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.