Question

In open interval \(\left(0, \frac{\pi}{2}\right)\),
(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

Ans: (B)
Hint \(: f(x)=\cos x+x \sin x-1\)
$$
\begin{aligned}
&\Rightarrow f^{\prime}(x)=-\sin x+\sin x+x \cos x>0 ; x \in\left(0, \frac{\pi}{2}\right) \\
&\Rightarrow f(x) \text { is increasing function } \\
&\Rightarrow f(x)>f(0) \\
&\cos x+x \sin x-1>0 \\
&\cos x+x \sin x>1
\end{aligned}
$$