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Limits Continuity and Differentiability question

2008 · Shift 1 · Q32

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JEE Advanced
Mathematics
Limits Continuity and Differentiability
2008 · Shift 1 · Q32

Limits Continuity and Differentiability question

2008 · Shift 1 · Q32

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMultiple correct+4 / −2

Let

f(x)

be a non-constant twice differentiable function defined on

\left( { - \infty ,\infty } \right)

such that

f\left( x \right) = f\left( {1 - x} \right)

and

f'\left( {{1 \over 4}} \right) = 0.

Then,

A
$f''\left( x \right)$ vanishes at least twice on $\left[ {0,1} \right]$
B
$f'\left( {{1 \over 2}} \right) = 0$
C
$\int\limits_{ - 1/2}^{1/2} {f\left( {x + {1 \over 2}} \right)\sin x\,dx} = 0$
D
$\int\limits_0^{1/2} {f\left( t \right){e^{\sin \,\pi t}}dt = } \int\limits_{1/2}^1 {f\left( {1 - t} \right){e^{\sin \,\pi t}}dt}$

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