1

Let $f\left(\frac xy\right)=\frac {f(x)}{f(y)}$ for all $x,y \in \mathbb{R}$ and suppose $f'(1)$ exists. The area under curve $f(x)$ bounded by the $x$-axis, $x=0$ and $x=1$ is $\frac 13$.

Find $$\lim_{n\to\infty }\sum_{i=0}^n e^\frac in f\left(\frac{\sqrt{i}}{n}\right)$$

This was a 12th standard test question. The answer key tells $1$.

STK
  • 73
  • I tried it but could not get any answer – STK Oct 27 '16 at 09:36
  • Show your working. Substitute $f(x)/f(y)$ for $f(\sqrt{i}/n)$ for starters. However, from your question it's not clear what $f(x/y)$ is equal to... what does $f(x)/f(y)\nabla x,y\in\mathfrak{R}$ mean? Perhaps this needs clarifying. – pshmath0 Oct 27 '16 at 09:41
  • That statement is valid for all real values of x and y – STK Oct 27 '16 at 09:42
  • Better to say "for all", or use $\forall$ then... and I would suggest using $\mathbb{R}$ instead of $\mathfrak{R}$... I've updated the question... – pshmath0 Oct 27 '16 at 09:43
  • What is $r$ in your statment ? – H.C. Lefevre Oct 27 '16 at 09:44
  • Sorry that was meant to be i – STK Oct 27 '16 at 09:51
  • I think the first information you give can tell us a lot about $f$, taking $x=0$, this looks like a Riemann sum – H.C. Lefevre Oct 27 '16 at 09:53
  • Have a look at this answer for solving the functional equation. What you have to evaluate looks like a Riemann sum and you are also given that $\int_0^1f(x),dx=\frac{1}{3}$. – StubbornAtom Oct 27 '16 at 10:30

0 Answers0