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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f(x)\, = \left\{ {\begin{array}{*{20}{c}}{x + 1,}&{{\rm{when}}}&{x < 2}\\{2x - 1,}&{{\rm{when}}}&{x \ge 2}\end{array}} \right.\,,\,$ then $f'(2) = $
  • A
    $0$
  • B
    $1$
  • C
    $2$
  • ✓
    Does not exist
✓

Answer

Correct option: D.
Does not exist
d
(d) $Rf'(2)$$ = \mathop {{\rm{lim}}}\limits_{h \to 0} \frac{{f(2 + h) - f(2)}}{h}$

$ = \mathop {{\rm{lim}}}\limits_{h \to 0} \frac{{2(2 + h) - 1 - (4 - 1)}}{h}$

$ = \mathop {{\rm{lim}}}\limits_{h \to 0} \frac{{4 + 2h - 1 - 3}}{h} = 2$

and $Lf'(2) = \mathop {{\rm{lim}}}\limits_{h \to 0} \frac{{f(2 - h) - f(2)}}{{ - h}} = \mathop {{\rm{lim}}}\limits_{h \to 0} \frac{{2 - h + 1 - 3}}{{ - h}} = 1$.

Thus $f'(2)$ does not exist.

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