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Let f(x) be a function satisfying the co...

Let `f(x)` be a function satisfying the condition `f(-x) = f(x)` for all real x. If `f'(0)` exists, then its value is equal to

Text Solution

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The correct Answer is:
`f'(0)=0`
Since, f(-x) = f(x)
`:. ` f(x) is an even function.
`:. F'(0) = underset( h to 0) lim (f(0+h)-f(0))/h `
` = underset( h to 0) lim (f(0-h)-f(0))/(-h) " "[:' f(-h) = f(h)]`
Since, f' (0) exists.
`:. " " R f'(0) = L f'(0) `
` rArr underset( h to 0) lim (f(h)-f(0))/h = underset( h to 0) lim (f(h)-f(0))/(-h) `
` rArr 2 underset( h to 0) lim (f(h) - f(0))/h = 0`
` rArr underset( h to 0) lim (f(h) - f(0))/ h = 0`
`:." " f'(0) =0 `
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