Prove inequality $x^sy^{1-s} \leq sx + (1-s)y$ Given $s \in (0,1)$, prove $$x^sy^{1-s} \leq sx + (1-s)y$$ for $x,y > 0$
Tried some algebraic manipulations but I'm guessing I need to use some trick. 
Any suggestions, hints?

Recall that the natural logarithm is a concave function. That is,
$$\log(sx + (1 - s)y) \ge s\log(x) + (1 - s)\log(y)$$
for all $x, y > 0$ and $s \in [0, 1]$. Using log laws,
$$\log(sx + (1 - s)y) \ge \log(x^s y^{1-s}).$$
Recall that the natural exponential is a monotone increasing function, meaning it respects inequalities. Taking the exponential of both sides yields the desired inequality.
