Question

Find the range of the function $f:[0,\,1]\to R,\,\,f(x)={{x}^{3}}-{{x}^{2}}+4x+2{{\sin }^{-1}}x$ ?

• A. $[-(\pi +2),\,0]$
• B. $[0,4+\pi ]$
• C. $[2,3]$
• D. $(0,\,2+\pi ]$

Answer 1

Answer: (B)

$[0,4+\pi ]$

Answer 2

We have $f:[0,\,1]\to R$ $f(x)={{x}^{3}}-{{x}^{2}}+4x+2{{\sin }^{-1}}x$ Now $f'(x)=3{{x}^{2}}-2x+4+\frac{2}{\sqrt{1-{{x}^{2}}}}$ For $x\,[0,\,\,1],\,\,\,f'\,(x)>0$ Hence, it is a increasing function at $x=0,\,f(0)=0-0+4(0)+2si{{n}^{-1}}(0)=0$ at $x=1,f(1)=1-1+4(1)+2{{\sin }^{-1}}(1)$
$=4+2\left( \frac{\pi }{2} \right)=4+\pi$
$\therefore$ Range of $f(x)\,[0,\,4+\pi ]$