Check sibling questions

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

(A)1Β 

(B) √3

(C) 2Β 

(D) None of these

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

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Transcript

Question 26 For the function f (x) = x + 1/π‘₯ , x ∈ [1, 3], the value of c for mean value theorem is 1 (B) √3 (C) 2 (D) None of these 𝑓(π‘₯)="x + " 1/π‘₯ in interval [1, 3] Checking conditions for Mean value Theorem 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 1 We need to check if 𝑓(π‘₯)="x + " 1/π‘₯ is continuous in the interval [1, 3] Let π’ˆ(𝒙)=𝒙 and 𝐑(𝒙)=𝟏/𝒙 We know that, π’ˆ(𝒙)=π‘₯ is continuous as it is a polynomial function And, 𝐑(𝒙)=1/π‘₯ is continuous for all π‘₯ except for 𝒙=𝟎 ∴ h(π‘₯)=1/π‘₯ is continuous in the interval [1, 3] Hence, 𝒇(𝒙)=π’ˆ(𝒙)+𝒉(𝒙) is also continuous in the interval [1, 3] Condition 2 We need to check if 𝑓(π‘₯)="x + " 1/π‘₯ is differentiable at ("1, 3") A function is said to be differentiable if the derivative of the function exists. Differentiating 𝑓(π‘₯) wrt π‘₯ 𝒇^β€² (𝒙)=πŸβˆ’πŸ/𝒙^𝟐 Since, derivative of the given function exists Hence, 𝒇(𝒙) is differentiable at ("1, 3") Since both conditions are satisfied From Mean Value Theorem, There exists a c ∈ (1, 3) such that, 𝑓^β€² (𝑐) = (𝑓(3) βˆ’ 𝑓(1))/(3 βˆ’1) πŸβˆ’πŸ/𝒄^𝟐 = ((πŸ‘ + 𝟏/πŸ‘) βˆ’ (𝟏 + 𝟏/𝟏))/𝟐 1βˆ’1/𝑐^2 = ((9 + 1)/3 βˆ’ 2)/2 1βˆ’1/𝑐^2 = (10/3 βˆ’ 2)/2 1βˆ’1/𝑐^2 = ((10 βˆ’ 6)/3)/2 1βˆ’1/𝑐^2 = 4/(3 Γ— 2 ) πŸβˆ’πŸ/𝒄^𝟐 = 2/3 πŸβˆ’πŸ/πŸ‘= 𝟏/𝒄^𝟐 (πŸ‘ βˆ’ 𝟐)/πŸ‘= 𝟏/𝒄^𝟐 𝟏/πŸ‘= 𝟏/𝒄^𝟐 𝑐^2=3 𝑐=±√3 𝒄=±𝟏.πŸ•πŸ‘ Either, c = βˆ’1.73 But, βˆ’1.73 βˆ‰(𝟏, πŸ‘) Or, c = 1.73 And, 1.73 ∈ (𝟏, πŸ‘) Therefore, 𝒄=βˆšπ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 12 years. He provides courses for Maths and Science at Teachoo.