For the function f (x) = x + 1/x , x ∈ [1, 3], the value of c for mean value theorem is

(A)1 

(B) √3

(C) 2 

(D) None of these

This question is similar to Example 43 - Chapter 5 Class 12 - Continuity and Differentiability

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  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Question 26 For the function f (x) = x + 1/๐‘ฅ , x โˆˆ [1, 3], the value of c for mean value theorem is 1 (B) โˆš3 (C) 2 (D) None of these ๐‘“(๐‘ฅ)="x + " 1/๐‘ฅ in interval [1, 3] Checking conditions for Mean value Theorem Conditions of Mean value theorem ๐‘“(๐‘ฅ) is continuous at {๐‘Ž, ๐‘} ๐‘“(๐‘ฅ) is differentiable at (๐‘Ž , ๐‘) If both conditions satisfied, then there exist some c in (๐‘Ž , ๐‘) such that ๐‘“โ€ฒ(๐‘) = (๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž))/(๐‘ โˆ’ ๐‘Ž) Condition 1 We need to check if ๐‘“(๐‘ฅ)="x + " 1/๐‘ฅ is continuous in the interval [1, 3] Let ๐’ˆ(๐’™)=๐’™ and ๐ก(๐’™)=๐Ÿ/๐’™ We know that, ๐’ˆ(๐’™)=๐‘ฅ is continuous as it is a polynomial function And, ๐ก(๐’™)=1/๐‘ฅ is continuous for all ๐‘ฅ except for ๐’™=๐ŸŽ โˆด h(๐‘ฅ)=1/๐‘ฅ is continuous in the interval [1, 3] Hence, ๐’‡(๐’™)=๐’ˆ(๐’™)+๐’‰(๐’™) is also continuous in the interval [1, 3] Condition 2 We need to check if ๐‘“(๐‘ฅ)="x + " 1/๐‘ฅ is differentiable at ("1, 3") A function is said to be differentiable if the derivative of the function exists. Differentiating ๐‘“(๐‘ฅ) wrt ๐‘ฅ ๐’‡^โ€ฒ (๐’™)=๐Ÿโˆ’๐Ÿ/๐’™^๐Ÿ Since, derivative of the given function exists Hence, ๐’‡(๐’™) is differentiable at ("1, 3") Since both conditions are satisfied From Mean Value Theorem, There exists a c โˆˆ (1, 3) such that, ๐‘“^โ€ฒ (๐‘) = (๐‘“(3) โˆ’ ๐‘“(1))/(3 โˆ’1) ๐Ÿโˆ’๐Ÿ/๐’„^๐Ÿ = ((๐Ÿ‘ + ๐Ÿ/๐Ÿ‘) โˆ’ (๐Ÿ + ๐Ÿ/๐Ÿ))/๐Ÿ 1โˆ’1/๐‘^2 = ((9 + 1)/3 โˆ’ 2)/2 1โˆ’1/๐‘^2 = (10/3 โˆ’ 2)/2 1โˆ’1/๐‘^2 = ((10 โˆ’ 6)/3)/2 1โˆ’1/๐‘^2 = 4/(3 ร— 2 ) ๐Ÿโˆ’๐Ÿ/๐’„^๐Ÿ = 2/3 ๐Ÿโˆ’๐Ÿ/๐Ÿ‘= ๐Ÿ/๐’„^๐Ÿ (๐Ÿ‘ โˆ’ ๐Ÿ)/๐Ÿ‘= ๐Ÿ/๐’„^๐Ÿ ๐Ÿ/๐Ÿ‘= ๐Ÿ/๐’„^๐Ÿ ๐‘^2=3 ๐‘=ยฑโˆš3 ๐’„=ยฑ๐Ÿ.๐Ÿ•๐Ÿ‘ Either, c = โˆ’1.73 But, โˆ’1.73 โˆ‰(๐Ÿ, ๐Ÿ‘) Or, c = 1.73 And, 1.73 โˆˆ (๐Ÿ, ๐Ÿ‘) Therefore, ๐’„=โˆš๐Ÿ‘ So, the correct answer is (B)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.