In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $

$$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$ Where, $$f(0)=f(\pi )=0$$

It then asked for the construction of functions in the use of approximate $\pi $. The question itself is not difficult at all, but I'm pretty interested in the reason why the inequality works. However, it seems like a fresh high school student is not eligible for it XD and I even got something non-sense.

So could anyone help me? Thanks a lot for any hint, guide, or most precisely, proof.


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