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Maths-

General

Easy

Question

Let $f left parenthesis x right parenthesis equals stack l i m with n rightwards arrow infinity below left square bracket fraction numerator n to the power of n left parenthesis x plus n right parenthesis left parenthesis x plus n over 2 right parenthesis..... left parenthesis x plus n over n right parenthesis over denominator n factorial left parenthesis x squared plus n squared right parenthesis left parenthesis x squared plus n squared over 4 right parenthesis times left parenthesis x squared plus n squared over n squared right parenthesis end fraction right square bracket to the power of x divided by n end exponent$ , for all $x greater than 0$ Then

$f (\frac{1}{2}) \geq f (1)$
$f (\frac{1}{3}) \leq f (\frac{2}{3})$
$f^{}$ (2)≤0'/>
$f^{}$ (3)f(3)≥f'(2)f(2)'/>

The correct answer is: $f to the power of ´ end exponent left parenthesis 2 right parenthesis less or equal than 0$

The given function is
$f left parenthesis x right parenthesis equals stack l i m with n rightwards arrow infinity below left square bracket fraction numerator n to the power of n end exponent left parenthesis x plus n right parenthesis left parenthesis x plus n divided by 2 right parenthesis times. times. times. left parenthesis x plus n divided by n right parenthesis over denominator n factorial left parenthesis x to the power of 2 end exponent plus n to the power of 2 end exponent right parenthesis left parenthesis x to the power of 2 end exponent plus n to the power of 2 end exponent divided by 4 right parenthesis left parenthesis x to the power of 2 end exponent plus n to the power of 2 end exponent divided by n to the power of 2 end exponent right parenthesis end fraction right square bracket to the power of x divided by n end exponent for all x greater than 0$ $rightwards double arrow f left parenthesis x right parenthesis equals stack l i m with n rightwards arrow infinity below left square bracket fraction numerator n to the power of n end exponent n to the power of n end exponent left parenthesis 1 plus fraction numerator chi over denominator n end fraction right parenthesis left parenthesis 1 plus fraction numerator 2 x over denominator n end fraction right parenthesis. times. times. times left parenthesis 1 plus fraction numerator n x over denominator n end fraction right parenthesis over denominator n factorial n factorial left parenthesis 1 plus fraction numerator chi to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis left parenthesis 1 plus fraction numerator 4 x to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis left parenthesis 1 plus fraction numerator n to the power of 2 end exponent x to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis end fraction cross times fraction numerator left parenthesis n factorial right parenthesis to the power of 2 end exponent over denominator left parenthesis n to the power of 2 end exponent right parenthesis to the power of n end exponent end fraction right square bracket to the power of chi divided by n end exponent$ $f left parenthesis x right parenthesis equals stack l i m with n rightwards arrow infinity below left square bracket fraction numerator left parenthesis 1 plus fraction numerator chi over denominator n end fraction right parenthesis left parenthesis 1 plus fraction numerator 2 x over denominator n end fraction right parenthesis times. times. times. left parenthesis 1 plus fraction numerator n x over denominator n end fraction right parenthesis over denominator left parenthesis 1 plus fraction numerator chi to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis left parenthesis 1 plus fraction numerator 4 x to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis left parenthesis 1 plus fraction numerator n to the power of 2 end exponent x to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction right parenthesis end fraction right square bracket to the power of x divided by n end exponent$ $f left parenthesis x right parenthesis equals x subscript n rightwards arrow infinity end subscript vertical line i m left square bracket not stretchy sum subscript r equals 1 end subscript superscript n end superscript fraction numerator 1 over denominator n end fraction$ lln $left square bracket fraction numerator 1 plus fraction numerator r x over denominator n end fraction over denominator 1 plus fraction numerator r to the power of 2 end exponent x to the power of 2 end exponent over denominator n to the power of 2 end exponent end fraction end fraction 1 right square bracket$
In $f left parenthesis x right parenthesis equals x not stretchy integral subscript 0 end subscript superscript 1 end superscript I n left parenthesis fraction numerator 1 plus x gamma over denominator 1 plus x to the power of 2 end exponent gamma to the power of 2 end exponent end fraction right parenthesis d gamma$
Substituting $x gamma equals ‡$ , we $g e t rightwards double arrow d gamma equals fraction numerator d r over denominator chi end fraction$ .Therefore,
In $f left parenthesis x right parenthesis equals fraction numerator chi over denominator chi end fraction not stretchy integral subscript 0 end subscript superscript chi end superscript vertical line n left parenthesis fraction numerator 1 plus ‡ over denominator 1 plus ‡ to the power of 2 end exponent end fraction right parenthesis d ‡ equals not stretchy integral subscript 0 end subscript superscript chi end superscript vertical line n left parenthesis fraction numerator 1 plus ‡ over denominator 1 plus ‡ to the power of 2 end exponent end fraction right parenthesis d ‡$
Applying Newton‐Leibniz rule, we get
$fraction numerator f to the power of ´ end exponent left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction equals vertical line n left parenthesis fraction numerator 1 plus x over denominator 1 plus x to the power of 2 end exponent end fraction right parenthesis equals f to the power of ´ end exponent left parenthesis x right parenthesis equals f left parenthesis x right parenthesis vertical line n left parenthesis fraction numerator 1 plus x over denominator 1 plus x to the power of 2 end exponent end fraction right parenthesis$
lt is obvious that $f left parenthesis x right parenthesis greater than 0 for all x greater than 0.$
For $x equals 2 colon fraction numerator f to the power of ´ end exponent left parenthesis 2 right parenthesis over denominator f left parenthesis 2 right parenthesis end fraction equals vertical line n left parenthesis fraction numerator 3 over denominator 5 end fraction right parenthesis less than 0 equals f to the power of ´ end exponent left parenthesis 2 right parenthesis less or equal than 0$
Hence, option (C) is correct.

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