QJEEAdv08Ma297e: JEE Advanced 2008 Mathematics - Class 12 JEEAdv08Ma297e Solution: Let f and g be real valued functions defined on interval ( − 1…

Question

Verified Solution • 2025–26 Session moderate

Let

f

and

g

be real valued functions defined on interval

(- 1, 1)

such that

g^{″} (x)

is continuous,

g (0) \neq 0.

g^{'} (0) = 0

,

g^{″} (0) \neq 0

, and

f (x) = g (x) \sin x

STATEMENT - 1: $lim_{x \to 0} [g (x) \cot x - g (0) \cos e c x] = f^{″} (0)$ and

STATEMENT - 2: $f^{'} (0) = g (0)$

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

Accepted Answer

$lim_{x \to 0} \frac{g (x) \cos x - g (0)}{\sin x}$

$= lim_{x \to 0} \frac{g^{'} (x) \cos x - g (x) \sin x}{\cos x}$ (Applying L-Hospital rule)

$g^{'} (0) - 0 = 0 = f^{″} (0)$

Statement - 1 is True.

$f^{'} (x) = g (x) \cos x + g^{'} (x) \sin x$

$f^{'} (0) = g (0)$

Statement - 2 is True.

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