Young's inequality for product can be stated as follows: $ab \leq \frac{1}{p}a^p + \frac{1}{q}b^q$ where a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p + 1/q = 1.

If we fix a and b, find p, q such that $f(p, q) = \frac{1}{p}a^p + \frac{1}{q}b^q$ is minimum.

Currently I am looking at the approach of equating the partial derivatives of f(p, q) w.r.t. q and p to zero, and find solutions for q and p. However, it looks like there is no closed-form solution.

Please kindly advise.


Hint: equality in Young's inequality is iff $a^p=b^q \iff a^x=b^{1-x}$ where $x=\dfrac1q$. Now can you find $x$, and hence $p, q$ ?

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  • $\begingroup$ Thanks. The solution is really unexpected :D Maybe I over-thought when trying to use calculus to solve the problem... $\endgroup$ – mr noname Jan 24 '15 at 14:36

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