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Definite Integration question

2016 · Shift 2 · Q25

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PYQ Library
JEE Advanced
Mathematics
Definite Integration
2016 · Shift 2 · Q25

Definite Integration question

2016 · Shift 2 · Q25

JEE AdvancedMathematicsDefinite IntegrationMultiple correct+4 / −2

Let

f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},

for all

x\gt 0.

Then

A
$f\left( {{1 \over 2}} \right) \ge f\left( 1 \right)$
B
$f\left( {{1 \over 3}} \right) \le f\left( {{2 \over 3}} \right)$
C
$\,f'\left( 2 \right) \le 0$
D
$\,{{f'\left( 3 \right)} \over {f\left( 3 \right)}} \ge {{f'\left( 2 \right)} \over {f\left( 2 \right)}}$

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