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

Transcript

Ex 5.8, 3 If ๐‘“ : [โ€“ 5, 5] โ†’ ๐‘ is a differentiable function and if ๐‘“ โ€ฒ(๐‘ฅ) does not vanish anywhere, then prove that ๐‘“ (โ€“5) โ‰  ๐‘“ (5). ๐‘“ : [โ€“ 5, 5] โ†’ ๐‘ is a differentiable โ‡’ We know that every differentiable function is continuous. Therefore f is continuous & differentiable both on (โˆ’5, 5) By Mean Value Theorem There exist some c in (5, โˆ’5) Such that ๐‘“^โ€ฒ (๐‘)=(๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž))/(๐‘ โˆ’ ๐‘Ž) Given that ๐‘“^โ€ฒ (๐‘ฅ) does not vanish any where โ‡’ ๐‘“^โ€ฒ (๐‘ฅ) โ‰  0 for any value of x Thus, ๐‘“^โ€ฒ (๐‘) โ‰  0 (๐‘“(5) โˆ’ ๐‘“(โˆ’5))/(5 โˆ’(โˆ’5) ) โ‰  0 (๐‘“(5) โˆ’ ๐‘“(โˆ’5))/(5 + 5) โ‰  0 ๐‘“(5)โˆ’ ๐‘“(โˆ’5) โ‰  0 ร— 10 ๐‘“(5)โˆ’ ๐‘“(โˆ’5) โ‰  0 ๐‘“(5) "โ‰ " ๐‘“(โˆ’5) Hence proved.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.