Question

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

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

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

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)=g\left(x\right)\sin x$ we have $f^{\prime }\left(x\right)=g\left(x\right)\cos x+g^{\prime }\left(x\right)\sin x$
$\therefore$ $f^{\prime }\left(0\right)=g\left(0\right)\cos 0+g^{\prime }\left(0\right)\sin =g\left(0\right)$

$f^{\prime \prime }\left(x\right)=2g^{\prime }\left(x\right)\cos x-g\left(x\right)\sin x+g^{\prime \prime }\left(x\right)\sin x$

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

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

(by using L-Hospital rule)

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