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Hi guys I have a doubt.

How can I prove that

(∫ (from 0 to pi) sin^7 xdx)(∫ (from 0 to pi) sin^(7/6) xdx)^6 is at most 128

But how can I prove that the lower bound of this expression is (pi/2)^7

I think is a very interesting and not an easy question so any ideas? A guidance or something... Thanks beforehand!

[Moderator edit]: The problem appears to be to prove the following inequalities:

$$ \left( \frac{ \pi}{2}\right)^{ \! \! 7} \le \int_{0}^{ \pi}\sin^{7}(x) \, dx \cdot \left( \int_{0}^{ \pi}\sin^{7/6}(x) \, dx\right)^{ \! \! 6} \le 128.$$

How can I prove that

(∫ (from 0 to pi) sin^7 xdx)(∫ (from 0 to pi) sin^(7/6) xdx)^6 is at most 128

But how can I prove that the lower bound of this expression is (pi/2)^7

I think is a very interesting and not an easy question so any ideas? A guidance or something... Thanks beforehand!

[Moderator edit]: The problem appears to be to prove the following inequalities:

$$ \left( \frac{ \pi}{2}\right)^{ \! \! 7} \le \int_{0}^{ \pi}\sin^{7}(x) \, dx \cdot \left( \int_{0}^{ \pi}\sin^{7/6}(x) \, dx\right)^{ \! \! 6} \le 128.$$

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