The function g (x) = [ . x + b, , , ,x < 0 x, , , ,x 0 can be mad… - Vidyadip

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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