Question

The range of $f\left(x\right)=-x^3+x^2-x+{\cos }^{-1}x$ is:

• A. $[-1,3+\pi ]$
• B. $[0,\pi -1]$
• C. $[-1,2+\pi ]$
• D. $[-1,\pi ]$

Answer 1

The correct option is A $[-1,3+\pi ]$

Given $f\left(x\right)=-x^3+x^2-x+{\cos }^{-1}x$

The domain for ${\cos }^{-1}x$ is $[-1,1]$. The range is $[0,\pi ]$

So when $x=-1$, $f\left(x\right)=f(-1)=-(-1{)}^3+(-1{)}^2-(-1)+\pi =1+1+1+\pi =3+\pi$

When $x=1$, $f\left(x\right)=f\left(1\right)=-(1{)}^3+(1{)}^2-\left(1\right)+0=-1+1-1=-1$

The range of $f\left(x\right)$ is $[-1,3+\pi ]$