Question

The range of the functions $f\left(x\right)={\int }_1^x|t|dt,xϵ[\frac{-1}{2},\frac{1}{2}]$ is

• A. $[\frac{3}{8},\frac{5}{8}]$
• B. $[\frac{-5}{8},\frac{3}{8}]$
• C. $[\frac{-3}{8},\frac{5}{8}]$
• D. $[\frac{-5}{8},\frac{-3}{8}]$

Answer 1

The correct option is D $[\frac{-5}{8},\frac{-3}{8}]$

If $0\le x\le \frac{1}{2}\Rightarrow f\left(x\right)={\int }_1^{x}tdt=\frac{x^2-1}{2}$
If -$\frac{1}{2}\le x\le 0\Rightarrow f\left(x\right)={\int }_1^{0}tdt-{\int }_0^{x}tdt=-\left(\frac{x^2+1}{2}\right)\Rightarrow f^1\left(x\right)>0\forall xϵ[\frac{-1}{2},\frac{1}{2}]$
$\therefore$ Range of f(x) is $\left[f\right(\frac{-1}{2}),f(\frac{1}{2}\left)\right]\Rightarrow [\frac{-5}{8},\frac{-3}{8}]$