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]