WBJEE · Maths · Application of Derivatives
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 & Solution
Correct Answer
(B) \(\cos x+x \sin x>1\)
Step-by-step Solution
Detailed explanation
Hint: \(f(x)=\cos x+x \sin x-1\) \(\Rightarrow f^{\prime}(x)=-\sin x+\sin x+x \cos x>0 ; x \in\left(0, \frac{\pi}{2}\right)\) \(\Rightarrow f(x)\) is increasing function \(\begin{array}{l} \Rightarrow f(x)>f(0) \\ \cos x+x \sin x-1>0 \\ \cos x+x \sin x>1 \end{array}\)
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