JEE Mains · Maths · STD 12 - 6. Application of derivatives
For the function \(f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right) \text {, where } x \in\left[0, \frac{\pi}{2}\right] \text {, }\) consider the following two statements : (\(I\)) \(\mathrm{f}\) is increasing in \(\left(0, \frac{\pi}{2}\right)\). (\(II\)) \(\mathrm{f}^{\prime}\) is decreasing in \(\left(0, \frac{\pi}{2}\right)\). Between the above two statements,
- A only (\(I\)) is true.
- B only (\(II\)) is true.
- C neither (\(I\)) nor (\(II\)) is true .
- D both (\(I\)) and (\(II\)) are true.
Answer & Solution
Correct Answer
(D) both (\(I\)) and (\(II\)) are true.
Step-by-step Solution
Detailed explanation
\( f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right) \quad x \in\left[0, \frac{\pi}{2}\right] \) \( \mathrm{f}^{\prime}(\mathrm{x})=\cos \mathrm{x}+3-\frac{2}{\pi}(2 \mathrm{x}+1)>0 \mathrm{f}(\mathrm{x}) \uparrow \) \( f^{\prime}(x)=-\sin x+0-\frac{\pi}{2}(2) \)…
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