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Ex 5.8, 3 - If f(x) is differentiable and f'(x) does not vanish

Ex 5.8, 3 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Ex 5.8, 3 If 𝑓 : [– 5, 5] β†’ 𝐑 is a differentiable function and if 𝑓 β€²(π‘₯) does not vanish anywhere, then prove that 𝑓 (–5) β‰  𝑓 (5). 𝑓 : [– 5, 5] β†’ 𝐑 is a differentiable β‡’ We know that every differentiable function is continuous. Therefore f is continuous & differentiable both on (βˆ’5, 5) By Mean Value Theorem There exist some c in (5, βˆ’5) Such that 𝑓^β€² (𝑐)=(𝑓(𝑏) βˆ’ 𝑓(π‘Ž))/(𝑏 βˆ’ π‘Ž) Given that 𝑓^β€² (π‘₯) does not vanish any where β‡’ 𝑓^β€² (π‘₯) β‰  0 for any value of x Thus, 𝑓^β€² (𝑐) β‰  0 (𝑓(5) βˆ’ 𝑓(βˆ’5))/(5 βˆ’(βˆ’5) ) β‰  0 (𝑓(5) βˆ’ 𝑓(βˆ’5))/(5 + 5) β‰  0 𝑓(5)βˆ’ 𝑓(βˆ’5) β‰  0 Γ— 10 𝑓(5)βˆ’ 𝑓(βˆ’5) β‰  0 𝑓(5) "β‰ " 𝑓(βˆ’5) Hence proved.

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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.