## 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

Last updated at Nov. 18, 2021 by Teachoo

This question is similar to

Example 43 - Chapter 5 Class 12 -Continuity and Differentiability

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Question 26 For the function f (x) = x + 1/๐ฅ , x โ [1, 3], the value of c for mean value theorem is 1 (B) โ3 (C) 2 (D) None of these ๐(๐ฅ)="x + " 1/๐ฅ in interval [1, 3] Checking conditions for Mean value Theorem Conditions of Mean value theorem ๐(๐ฅ) is continuous at {๐, ๐} ๐(๐ฅ) is differentiable at (๐ , ๐) If both conditions satisfied, then there exist some c in (๐ , ๐) such that ๐โฒ(๐) = (๐(๐) โ ๐(๐))/(๐ โ ๐) Condition 1 We need to check if ๐(๐ฅ)="x + " 1/๐ฅ is continuous in the interval [1, 3] Let ๐(๐)=๐ and ๐ก(๐)=๐/๐ We know that, ๐(๐)=๐ฅ is continuous as it is a polynomial function And, ๐ก(๐)=1/๐ฅ is continuous for all ๐ฅ except for ๐=๐ โด h(๐ฅ)=1/๐ฅ is continuous in the interval [1, 3] Hence, ๐(๐)=๐(๐)+๐(๐) is also continuous in the interval [1, 3] Condition 2 We need to check if ๐(๐ฅ)="x + " 1/๐ฅ is differentiable at ("1, 3") A function is said to be differentiable if the derivative of the function exists. Differentiating ๐(๐ฅ) wrt ๐ฅ ๐^โฒ (๐)=๐โ๐/๐^๐ 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 โ (1, 3) such that, ๐^โฒ (๐) = (๐(3) โ ๐(1))/(3 โ1) ๐โ๐/๐^๐ = ((๐ + ๐/๐) โ (๐ + ๐/๐))/๐ 1โ1/๐^2 = ((9 + 1)/3 โ 2)/2 1โ1/๐^2 = (10/3 โ 2)/2 1โ1/๐^2 = ((10 โ 6)/3)/2 1โ1/๐^2 = 4/(3 ร 2 ) ๐โ๐/๐^๐ = 2/3 ๐โ๐/๐= ๐/๐^๐ (๐ โ ๐)/๐= ๐/๐^๐ ๐/๐= ๐/๐^๐ ๐^2=3 ๐=ยฑโ3 ๐=ยฑ๐.๐๐ Either, c = โ1.73 But, โ1.73 โ(๐, ๐) Or, c = 1.73 And, 1.73 โ (๐, ๐) Therefore, ๐=โ๐ So, the correct answer is (B)

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Chapter 5 Class 12 Continuity and Differentiability (Term 1)

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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.