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प्रश्न
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
विकल्प
`3/2`
1
0
–1
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उत्तर
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is 1.
Explanation:
Given f(x) = x – [x]
We have to first check for differentiability of f(x) at x = `1/2`
∴ Lf'`(1/2)` = L.H.D
= `lim_(h -> 0) (f[1/2 - h] - f[1/2])/(-h)`
= `lim_(h -> 0) ((1/2 - h) - [1/2 - h] - 1/2 + [1/2])/(-h)`
= `lim_(h -> 0) (1/2 - h - 0 - 1/2 + 0)/(-h)`
= `(-h)/(-h)`
= 1
Rf'`(1/2)` = R.H.D
= `lim_(h -> 0) (f(1/2 + h) - f(1/2))/h`
= `lim_(h -> 0) ((1/2 + h) - [1/2 + h] - 1/2 + [1/2])/h`
= `lim_(h -> 0) (1/2 + h - 1 - 1/2 + 1)/h`
= `h/h`
= 1
Since L.H.D = R.H.D
∴ f'`(1/2)` = 1
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