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STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
The function $g (x) =$ $\left[ \begin{gathered}   \hfill \\   \hfill \\ \end{gathered}  \right.$$\begin{gathered}  x + b,\,\,\,x < 0 \hfill \\   \hfill \\  \cos x,\,\,\,x \geqslant 0 \hfill \\ \end{gathered} $ can be made differentiable at $x = 0.$
  • A
    if $b$ is equal to zero
  • B
    if $b$ is not equal to zero
  • C
    if $b$ takes any real value
  • ✓
    for no value of $b$
✓

Answer

Correct option: D.
for no value of $b$
d
$g’ (0^+) =$$\mathop {Lim}\limits_{h \to 0} \,\frac{{\cosh  - 1}}{h}$ $= 0$
$g ‘ (0^-) =$$\mathop {Lim}\limits_{h \to 0} \,\frac{{ - h + b - 1}}{{ - h}}$ for existence of line $b = 1$ thus $g ‘ (0^-) = 1$
Hence $g$ can not be made differentiable for any value of $b$.

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