Check sibling questions

The value of c in Rolle’s Theorem for the function f (x) = e^x sin x,

x∈ [0, π] is

(A) π/6  (B) π/4 

(C) π/2 

(D) 3π/4

This question is similar to Ex 5.8, 1 - Chapter 5 Class 12 - Continuity and Differentiability


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Question 1 The value of c in Rolle’s Theorem for the function f (x) = 𝑒^π‘₯ sin π‘₯, π‘₯∈ [0, πœ‹] is (A) πœ‹/6 (B) πœ‹/4 (C) πœ‹/2 (D) 3πœ‹/4 𝑓 (π‘₯)= 𝑒^π‘₯ sin⁑〖π‘₯, π‘₯ ∈ [0,πœ‹] γ€— First, we will check if the conditions of Rolle’s theorem are satisfied Condition 1 We need to check if 𝑓(π‘₯)=𝑒^π‘₯ sin⁑〖π‘₯ γ€—is continuous at [𝟎,𝝅] Let π’ˆ(𝒙)=𝒆𝒙 and 𝒉(𝒙)=𝐬𝐒𝐧⁑𝒙 We know that, 𝑔(π‘₯)=𝑒^π‘₯ is continuous as it is an exponential function β„Ž(π‘₯)=sin⁑π‘₯ is continous as sin is continuous at [𝟎,𝝅] Hence, 𝒇(𝒙)=π’ˆ(𝒙)𝒉(𝒙) is also continuous at [0,πœ‹] ∴ 𝒇(𝒙)=𝑒^π‘₯ sin⁑π‘₯ is continuous at π‘₯∈[0,πœ‹] Condition 2 We need to check if 𝑓(π‘₯)=𝑒^π‘₯ sin⁑π‘₯ is differentiable at (𝟎,𝝅) A function is said to be differentiable if the derivative of the function exists. Differentiating 𝑓(π‘₯) wrt π‘₯ 𝒇^β€² (𝒙)=𝒆^𝒙 π’”π’Šπ’β‘π’™+𝒆^𝒙 πœπ¨π¬β‘π’™ Since, derivative of the given function exists Hence, 𝒇(𝒙) is differentiable at (0,πœ‹) Finding 𝒇(𝟎) 𝑓(π‘₯)=𝑒^π‘₯ sin⁑π‘₯ 𝑓(𝟎) = 𝑒^0 sin⁑0 = 𝑒^0 (0) = 0 Finding𝒇(𝝅) 𝑓(π‘₯) = 𝑒^π‘₯ sin⁑π‘₯ 𝑓(𝝅) = 𝑒^πœ‹ sinβ‘πœ‹ = 𝑒^πœ‹ (0) = 0 Condition 3 Hence 𝒇(𝟎) = 𝒇(𝝅) Now, 𝑓(π‘₯) = 𝑒^π‘₯ sin⁑π‘₯ 𝑓^β€² (π‘₯) =𝑒^π‘₯ sin⁑π‘₯+𝑒^π‘₯ cos⁑π‘₯ 𝑓^β€² (π‘₯) =𝑒^π‘₯ γ€–(sin〗⁑π‘₯+cos⁑π‘₯) 𝒇^β€² (𝒄) = 𝒆^𝒄 γ€–(π’”π’Šπ’γ€—β‘π’„+𝒄𝒐𝒔⁑𝒄) Since all three condition satisfied 𝑓^β€² (𝑐) = 0 𝒆^𝒄 γ€–(π’”π’Šπ’γ€—β‘π’„+𝒄𝒐𝒔⁑𝒄) = 𝟎 Either, 𝒆^𝒄=𝟎 It is not possible, since it is an exponential function. Or, γ€–(π’”π’Šπ’γ€—β‘π’„+𝒄𝒐𝒔⁑𝒄)=𝟎 sin⁑c=βˆ’cos⁑c Dividing by 𝒄𝒐𝒔⁑𝒄 on both sides sin⁑c/cos⁑c =βˆ’cos⁑c/cos⁑c π­πšπ§β‘γ€–πœ=βˆ’πŸγ€— c=3Ο€/4, 7Ο€/4, … Since, c ∈(0,Ο€) ∴ c = πŸ‘π…/πŸ’ So, the correct answer is (D)

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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.