<s>[INST] Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$  
I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$
thanks for helpers
 [/INST] There is no need for derivatives. If $x\in (0,1)$, then $1 - x > 0$ and $\sin^2 x\ge 0$, thus $(1 - x)\sin^2 x \ge 0$. Consequently, $x \le x + (1 - x)\sin^2 x$. Since $0 \le \sin^2 x \le 1$ and $1 - x > 0$, $(1 - x)\sin^2 x \le 1 - x$. Hence $x + (1 - x)\sin^2 x \le x + (1 - x) = 1$. 
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