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

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  1. Chapter 5 Class 12 Continuity and Differentiability
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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‘“๏ทฎโ€ฒ๏ทฏ ๐‘๏ทฏ= ๐‘“ ๐‘๏ทฏ โˆ’ ๐‘“ ๐‘Ž๏ทฏ๏ทฎ๐‘ โˆ’ ๐‘Ž๏ทฏ Given that ๐‘“๏ทฎโ€ฒ๏ทฏ ๐‘ฅ๏ทฏ does not vanish any where โ‡’ ๐‘“๏ทฎโ€ฒ๏ทฏ ๐‘ฅ๏ทฏ โ‰  0 for any value of x Thus, ๐‘“๏ทฎโ€ฒ๏ทฏ ๐‘๏ทฏ โ‰  0 ๐‘“ 5๏ทฏ โˆ’ ๐‘“ โˆ’5๏ทฏ๏ทฎ5 โˆ’ โˆ’5๏ทฏ๏ทฏ โ‰  0 ๐‘“ 5๏ทฏ โˆ’ ๐‘“ โˆ’5๏ทฏ๏ทฎ5 + 5๏ทฏ โ‰  0 ๐‘“ 5๏ทฏ โˆ’ ๐‘“ โˆ’5๏ทฏ โ‰  0 ร— 10 ๐‘“ 5๏ทฏ โˆ’ ๐‘“ โˆ’5๏ทฏ โ‰  0 ๐‘“ 5๏ทฏ โ‰  ๐‘“ โˆ’5๏ทฏ Hence proved.

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