This is a question that I came across today:  

let $x\in (0,1)$, and $k$ be postive intgers,such $k\ge 2$,


>I  conjecture following inequality maybe hold?
$$x\cdot \dfrac{(1-x)^{k-1}}{(k+1)^{k-2}}+\dfrac{(1-2x)^k}{k^k}\le \dfrac{1}{(k+2)^{k-1}}$$
This is my attempt

when $k=2$,then inequality  can be written as
$$x\cdot (1-x)+\dfrac{(1-2x)^2}{4}\le\dfrac{1}{4}$$
 it is obviously true.

when $k=3$then inequality can be written as
$$\dfrac{x(1-x)^2}{4}+\dfrac{(1-2x)^3}{27}\le\dfrac{1}{25}$$
or $$-\dfrac{(5x-1)^2(5x+8)}{2700}\le 0$$ it is clearly true.

and I have solve when $k=4,5$ it is hold,But I can't prove for any $k\ge 6$.Thanks