Let f:0, π→R be a twice differentiable function such that limt→xfxsint-ftsinxt-x=sin2x for all x∈0, π. If fπ6=-π12, then which of the following statement(s) is (are) TRUE ?

(C) There exists α∈0, π such that f′α=0

JEE Advanced · Mathematics · 29. Differential Eqns

Let $f : (0, π) \to R$ be a twice differentiable function such that $\lim_{t \to x} \frac{f (x) \sin t - f (t) \sin x}{t - x} = \sin^{2} x$ for all $x \in (0, π)$ . If $f (\frac{π}{6}) = - \frac{π}{12}$ , then which of the following statement(s) is (are) TRUE ?

A $f (\frac{π}{4}) = \frac{π}{4 \sqrt{2}}$
B $f (x) < \frac{x^{4}}{6} - x^{2}$ for all $x \in (0, π)$
C There exists $α \in (0, π)$ such that $f^{'} (α) = 0$
D $f^{"} (\frac{π}{2}) + f (\frac{π}{2}) = 0$

Verified Solution

Answer & Solution

Correct Answer

(B) $f (x) < \frac{x^{4}}{6} - x^{2}$ for all $x \in (0, π)$

Step-by-step Solution

Detailed explanation

\lim_{t \to x} \frac{f (x) \sin t - f (t) \sin x}{t - x} = \sin^{2} x

By using L'Hopital rule

\lim_{t \to x} \frac{f (x) \cos t - f^{'} (t) \sin x}{1} = \sin^{2} x

\Rightarrow f (x) \cos x - f^{'} (x) \sin x = \sin^{2} x

\Rightarrow - (\frac{f^{'} (x) \sin x - f (x) \cos x}{\sin^{2} x}) = 1

\Rightarrow - \frac{d \frac{f (x)}{\sin x}}{d x} = 1

\Rightarrow \frac{f (x)}{\sin x} = - x + c

Put

x = \frac{π}{6}

and

f (\frac{π}{6}) = - \frac{π}{12}

∴

c = 0 \Rightarrow f (x) = - x \sin x

(A)

f (\frac{π}{4}) = - \frac{π}{4} \frac{1}{\sqrt{2}}

(B)

f (x) = - x \sin x

\sin x > x - \frac{x^{3}}{6}

- x \sin x < - x^{2} + \frac{x^{4}}{6}

∴ f (x) < - x^{2} + \frac{x^{4}}{6} \forall x \in (0, π)

(C)

f^{'} (x) = - \sin x - x \cos x

f^{'} (x) = 0 \Rightarrow \tan x = - x \Rightarrow

there exist

α \in (0, π)

for which

f^{'} (α) = 0

(D)

f^{"} (x) = - 2 \cos x + x \sin x

f^{"} (\frac{π}{2}) = \frac{π}{2}, f (\frac{π}{2}) = - \frac{π}{2}

f^{"} (\frac{π}{2}) + f (\frac{π}{2}) = 0

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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