Question

If $f(x) = \prod\limits_{n=1}^{100}(x-n)^{n(101-n)}$ then find $\dfrac{f(101)}{f'(101)}$.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: f(x) = \prod\limits_{n=1}^{100}(x-n)^{n(101-n)}$

On taking log on both sides, we get

$\rm \:log f(x) =log\bigg( \prod\limits_{n=1}^{100}(x-n)^{n(101-n)}\bigg)$

$\rm \:log f(x) =log\bigg((x-1)^{1(101-1)}(x-2)^{2(101-2)} - - (x-100)^{100(101-100)}\bigg)$

$\rm \:log f(x) =log(x-1)^{1(101-1)} + log(x-2)^{2(101-2)} + - - + log(x-100)^{100(101-100)}$

can be further rewritten as

$\rm \: logf(x) = \sum\limits_{n=1}^{100}log\bigg((x-n)^{n(101-n)}\bigg)$

$\rm \: logf(x) = \sum\limits_{n=1}^{100}n(101 - n)log(x-n)$

On differentiating both sides w. r. t. x, we get

$\rm \: \dfrac{d}{dx} logf(x) =\dfrac{d}{dx} \sum\limits_{n=1}^{100}n(101 - n)log(x-n)$

$\rm \: \dfrac{f'(x)}{f(x)} = \sum\limits_{n=1}^{100}n(101 - n)\dfrac{d}{dx}log(x-n)$

$\rm \: \dfrac{f'(x)}{f(x)} = \sum\limits_{n=1}^{100}n(101 - n) \times \dfrac{1}{x - n}$

On substituting x = 101, we get

$\rm \: \dfrac{f'(101)}{f(101)} = \sum\limits_{n=1}^{100}n(101 - n) \times \dfrac{1}{101 - n}$

$\rm \: \dfrac{f'(101)}{f(101)} = \sum\limits_{n=1}^{100}n$

$\rm \: \dfrac{f'(101)}{f(101)} = \dfrac{100(100 + 1)}{2}$

$\rm \: \dfrac{f'(101)}{f(101)} = \dfrac{100(101)}{2}$

$\rm \: \dfrac{f'(101)}{f(101)} = 101 \times 50$

$\rm \: \dfrac{f'(101)}{f(101)} = 5050$

$\rm\implies \:\rm \: \dfrac{f(101)}{f'(101)} = \dfrac{1}{5050}$

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Formulae Used :-

$\boxed{\sf{  \:\rm \: log {x}^{y} = ylogx \: \: }}$

$\boxed{\sf{  \:\rm \: log(xy) = logx + logy \: \: }}$

$\boxed{\sf{  \:\rm \: \dfrac{d}{dx}logx = \frac{1}{x} \: \: }}$

$\boxed{\sf{  \:\rm \: \sum\limits_{k=1}^{n} \: k \: = \: \frac{n(n + 1)}{2} \: \: }}$

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Additional Information :-

$\boxed{\sf{  \:\rm \: \sum\limits_{k=1}^{n} \: {k}^{2} \: = \: \frac{n(n + 1)(2n + 1)}{6} \: \: }}$

$\boxed{\sf{  \:\rm \: \sum\limits_{k=1}^{n} \: {k}^{3} \: = \: \bigg(\frac{n(n + 1)}{2}\bigg)^{2} \: \: }}$

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$