HCU MCA 2013 Paper

Question : 48 of 75

Marks: +1, -0

If [x] denotes the largest integer ≤ x and if f(x) = x [x]. f ' (x) (where ever it exists) is given by?

Solution:

(d) Given curve;
f(x) = x[x]
Now ,Lf'(0)

= \lim ↙{ h → 0 } { f ( 0 - h ) - f ( 0 ) }/ { - h }

= \lim ↙ { h → 0 } { ( - h ) [ - h ] - 0 } /{ - h }

= \lim ↙ { h → 0 } { - h ( - 1 ) - 0 }/ { - h } = \lim ↙{ h → 0 } {h/{-h}}

= - 1

Rf'(0)

= \lim ↙{ h → 0 } { f ( 0 + h ) - f ( 0 ) }/ { h }

= \lim ↙ { h → 0 } { ( h ) [ h ] - 0 } /{ h }

= \lim ↙ { h → 0 } { h ×0 - 0 }/ { h } = \lim ↙{ h → 0 } {0/{h}}

\lim↙{h→0}0=0

∴ Lf'(0) ≠ Rf'(0)
Hence, f' (x) exists nowhere.

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