I'm trying to understand a proof of a Lemma that's in relation to Young's inequality, the lemma is the following : Suppose $\frac{1}{p} + \frac{1}{q} = 1$
Let $a,b>0$ and let $1<p<\infty$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $$ab = \min_{1<t<\infty} \frac{t^pa^p}{p} + \frac{t^{-q}b^q}{q}$$
It says this can be proved by "the methods of elementary calculus," but I don't see how this is the case. If you differentiate with respect to t the function that is being minimised you get $t^{p-1}a^p - t^{-q-1}b^q = 0$. But I don't understand how you get the proof from this.