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Let f(x) = x – [x]; ∈ R, then f'(12) is ______. - Mathematics

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Question

Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.

Options

  • `3/2`

  • 1

  • 0

  • –1

MCQ
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Solution

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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Chapter 13: Limits and Derivatives - Exercise [Page 244]

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NCERT Exemplar Mathematics [English] Class 11
Chapter 13 Limits and Derivatives
Exercise | Q 67 | Page 244

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