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)

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 ] $