Question

The values of $a$ for which the integral ${\int }_0^2|x-a|dx\ge 1$ is satisfied is/are

• A. $[2,\infty )$
• B. $(-\infty ,0]$
• C. $(0,2)$
• D. None of the above

Answer 1

Answer: (A), (B), (C)

$[2,\infty )$
$(-\infty ,0]$
$(0,2)$

For $a\le 0,$ given equation becomes
${\int }_0^2(x-a)dx\ge 1\Rightarrow a\le \frac{1}{2}\Rightarrow a\le 0$
For $0<a<2,$
${\int }_0^2|x-a|dx\ge 1\Rightarrow {\int }_0^a(a-x)dx+{\int }_a^2(x-a)dx\ge 1$
$\Rightarrow \frac{a^2}{2}+2-2a+\frac{a^2}{2}\ge 1\Rightarrow a^2-2a+1\ge 0\Rightarrow (a-1{)}^2\ge 0$
For $a\ge 2$,
${\int }_0^2|x-a|dx\ge 1\Rightarrow {\int }_0^2(a-x)dx\ge 1\Rightarrow 2a-2\ge 1\Rightarrow a\ge \frac{3}{2}\Rightarrow a\ge 2$