Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$ I have no idea how to do this. Any help would be appreciated. The chapter I'm on is about differentiation and the mean value theorem.
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$

This is a special case of geometric average vs arithmetic average.
if $\alpha$ is rational, say $\frac{p}{q}$, the first term is the geometric average of $p$ times $a$ and $(q-p)$ times $b$, and the secon term is the arithmetic average.
If $\alpha$ is not rational, then just use convergence.
