A) (a) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1
B) (b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
C) (c) Statement-1 is true, statement-2 is false.
D) (d) Statement-1 is false, Statement-2 is true
Correct Answer: C
Solution :
\[g(f(x))=\sin (f(x))=\left\{ \begin{matrix} \sin {{x}^{2}}, & x\ge 0 \\ -2x\,\cos {{x}^{2}}, & x<0 \\ \end{matrix} \right.\] \[(g(f(x)))'=\left\{ \begin{matrix} 2x\,\cos {{x}^{2}}, & x\ge 0 \\ -2x\,\cos {{x}^{2}}, & x<0 \\ \end{matrix} \right.\] R.H.D. of\[(g(f(0)))'=\underset{h\to {{0}^{+}}}{\mathop{\lim }}\,\frac{2h{{\cosh }^{2}}}{h}=2\] L.H.D. of\[(g(f(0)))'=\underset{h\to {{0}^{+}}}{\mathop{\lim }}\,\frac{2h{{\cosh }^{2}}}{-h}=-2\] Clearly gof is twice differentiable at\[x=0\]hence it is differentiable at\[x=0\]and its derivative is continuous at\[x=0\].You need to login to perform this action.
You will be redirected in
3 sec