Question

In an open interval $(0,\frac{\pi }{2})$,

• A. $\cos x+x\sin x<1$
• B. $\cos x+x\sin x>1$
• C. no specific order relation can be ascertained between $\cos x+x\sin x$ and $1$
• D. $\cos x+x\sin x<\frac{1}{2}$

Answer 1

The correct option is B $\cos x+x\sin x>1$

Let $\cos x+x\sin x=f\left(x\right)$
$f^{\prime }\left(x\right)=-\sin x+\sin x+x\cos x\Rightarrow f^{\prime }\left(x\right)=x\cos x>0\forall x\in (0,\frac{\pi }{2})\Rightarrow x>0\Rightarrow f\left(x\right)>f\left(0\right)\Rightarrow \cos x+x\sin x>1$