# Finding the lowest upper bound of product of two number using Young's inequality

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.

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 $p, q$ ?