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

Question

For the function

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

consider the following two statements :

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

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

Between the above two statements,

Accepted Answer

$\begin{gathered} f(x)=\sin x+3x-\frac{2}{\pi }({x}^{2}+x),x\in [0,\frac{\pi }{2}] \ {f}^{'}(x)=\cos x+3-\frac{2}{\pi }(2x+1) \ =\cos x-\frac{4x}{\pi }-\frac{2}{\pi }+3 \ x\in [0,\frac{\pi }{2}] \ \frac{4x}{\pi }\in [0,2] \end{gathered}$

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

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

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

$\Rightarrow f(x)$ is increasing

Now, ${f}^{''}(x)=-\sin x-\frac{4}{\pi }Hence, ${f}\^{'}(x)$ is decreasing

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Solutions · 1