For the function [f(x)= x+3 x-(2)/()(x^2+x), where x [0, ()/(2)],] consider the following …

Question

For the function

$$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$

consider the following two statements :

(I) $f$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$.

(II) $$f^{\prime}$$ is decreasing in $$\left(0, \frac{\pi}{2}\right)$$.

Between the above two statements,

Accepted Answer

$$\begin{aligned}
& f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right] \
& f^{\prime}(x)=\cos x+3-\frac{2}{\pi}(2 x+1) \
& =\cos x-\frac{4 x}{\pi}-\frac{2}{\pi}+3 \
& \text { as } x \in\left[0, \frac{\pi}{2}\right] \
& \frac{4 x}{\pi} \in[0,2]
\end{aligned}$$

$$\Rightarrow 3-\frac{2}{\pi}-\frac{4 x}{\pi}>0$$

and also $$\cos x>0$$ when $$x \in\left[0, \frac{\pi}{2}\right]$$

$$\Rightarrow f^{\prime}(x)>0$$

$$\Rightarrow f(x)$$

Solutions · 1