A) \[I<\frac{2}{3}\] and \[J<2\]
B) \[I>\frac{2}{3}\] and \[J<2\]
C) \[I>\frac{2}{3}\] and \[J>2\]
D) \[I<\frac{2}{3}\] and \[J>2\]
Correct Answer: D
Solution :
We know that \[\frac{\sin x}{x}<1\], when\[x\in \left( 0,1 \right)\Rightarrow \frac{\sin x}{\sqrt{x}}<\sqrt{x}\Rightarrow \int\limits_{0}^{1}{\frac{\sin x}{\sqrt{x}}<\frac{2}{3}}\] Again \[\frac{\cos x}{\sqrt{x}}<\frac{1}{\sqrt{x}}\] when \[x\in \left( 0,1 \right)\Rightarrow \int\limits_{0}^{1}{\frac{\cos x}{\sqrt{x}}<2}\]
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