JEE Advanced 2016 · Paper 2 · Mathematics Q43

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MathematicsMultiple correct+4 / -2Question 43 of 54

Let $f(x)=\displaystyle\lim_{n\to\infty}\left(\frac{n^n(x+n)\left(x+\frac n2\right)\cdots\left(x+\frac nn\right)}{n!\,(x^2+n^2)\left(x^2+\frac{n^2}{4}\right)\cdots\left(x^2+\frac{n^2}{n^2}\right)}\right)^{x/n}$ , for all $x>0$ . Then

f\left(\frac12\right)\ge f(1)

f\left(\frac13\right)\le f\left(\frac23\right)

f'(2)\le0

\dfrac{f'(3)}{f(3)}\ge\dfrac{f'(2)}{f(2)}

Answer and worked solution

Official answer: B, C

Take logs. For each $r$ , the factor is $\dfrac{n}{r}\cdot\dfrac{x+n/r}{x^2+n^2/r^2}=\dfrac{1+x(r/n)}{1+x^2(r/n)^2}$ . As a Riemann sum,
$\ln f(x)=x\int_0^1\ln\frac{1+xt}{1+x^2t^2}\,dt=\int_0^x\ln\frac{1+u}{1+u^2}\,du.$

Derivative.
$\frac{f'(x)}{f(x)}=\ln\frac{1+x}{1+x^2}.$
This is positive for $x<1$ and negative for $x>1$ .

(A) $f$ increases on $(0,1)$ , so $f(\frac12)<f(1)$ . False.
(B) $f(\frac13)<f(\frac23)$ . True.
(C) $f'(2)$ has the sign of $\ln\frac35<0$ . True.
(D) $\ln\frac4{10}<\ln\frac35$ . False.

(B), (C)

Topics:Limit of a sum Monotonicity

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JEE Advanced 2016 · Paper 2, Mathematics question 43