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 $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$

Hint: Notice that if we divide by $b$ the inequality is equivalent to $$(a/b)^\alpha \leq \alpha \cdot (a/b) + (1-\alpha)$$
set $t = a/b > 0$...
