Question

Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g^{\prime \prime }\left(x\right)$ is continuous, $g\left(0\right)\ne 0,g^{\prime }\left(0\right)=0,g^{\prime \prime }\left(0\right)\ne 0$, and $f\left(x\right)=g\left(x\right)\sin x.$

STATEMENT -1 : $\underset{x\to 0}{lim}\left[g\right(x)\cot x-g(0\left)cosecx\right]=f^{\prime \prime }\left(0\right)$
STATEMENT-2: $f^{\prime }\left(0\right)=g\left(0\right)$ .

• A. Statement-1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1
• B. Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

Answer: (B)

Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

$f^{\prime }\left(x\right)=g\left(x\right)cosx+\sin x.g^{\prime }\left(x\right)$
$\Rightarrow f^{\prime }\left(0\right)=g\left(0\right)$
$f^{\prime \prime }\left(x\right)=2g^{\prime }\left(x\right)\cos x-g\left(x\right)\sin x+\sin x{g}^{\prime \prime }\left(x\right)$
$\Rightarrow f^{\prime \prime }\left(0\right)=2g^{\prime }\left(0\right)=0$
But $\underset{x\to 0}{lim}\left[g\right(x)\cot x-g(0)cosecx]=\underset{x\to 0}{lim}\frac{g\left(x\right)\cos x-g\left(0\right)}{\sin x}$

$=\underset{x\to 0}{lim}\frac{g^{\prime }\left(x\right)\cos x-g\left(x\right)\sin x}{\cos x}$

$=g^{\prime }\left(0\right)=0=f^{\prime \prime }\left(0\right)$