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Limits Continuity and Differentiability question

2018 · Shift 2 · Q24

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JEE Advanced
Mathematics
Limits Continuity and Differentiability
2018 · Shift 2 · Q24

Limits Continuity and Differentiability question

2018 · Shift 2 · Q24

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMultiple correct+4 / −1

Let f : (0,

\pi

)

\to

R be a twice differentiable function such that

\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x

for all x

\in

(0,

\pi

). If

f\left( {{\pi \over 6}} \right) = - {\pi \over {12}}

, then which of the following statement(s) is (are) TRUE?

A
$f\left( {{\pi \over 4}} \right) = {\pi \over {4\sqrt 2 }}$
B
$f(x) \lt {{{x^4}} \over 6} - {x^2}$ for all x $\in$ (0, $\pi$ )
C
There exists $\alpha \in$ (0, $\pi$ ) such that f'( $\alpha$ ) = 0
D
$f''\left( {{\pi \over 2}} \right) + f\left( {{\pi \over 2}} \right) = 0$

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