# 2014 II 2)

Hi!

I'm having a lot of trouble understanding the solutions on the internet (e.g. MEI website/TSR) for the second part of i) "Show by means of counter example ...."

I don't really understand where do you get f(x) = 1-x/n from?

I would appreciate it a lot if anyone could explain this part of the question to me

:)

### You just want to come up with

You just want to come up with one example (there are many different possible ones) to show that if you have a function $f(x)$ such that not both of $f(0) = 0$ and $f(\pi) = 0$ are satisfied at once (i.e: at least one of $f(0),f(\pi)$ is $\neq 0$) then the result $(*)$ is not true.

One obvious function satisfying $f(0) = 0$ and $f(\pi) \neq 0$ is $f(x) = x$. Can you check that this is a counterexample? i.e: is the statement $\int_0^{\pi} x^2 \, \mathrm{d}x \leq \int_0^{\pi} 1^2 \, \mathrm{d}x$ true or false? If it's false, you've found a counterexample.

Can you find a counterexample in the case where $f(0) \neq 0$ and $f(\pi) = 0$? (hint: it's a small modification to the above function).

[As for $f = 1 - \frac{x}{n}$, this seems unnecessarily complicated]

### This clarified a lot, i was

This clarified a lot, i was still thinking in terms of sinnx

Thanks!!

### Awesome!

Awesome!

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