Question

If the range of the function f (x) $=6|x-1|+|3x-2|+2x,$ is $[a,\infty ]$ then the value of $a$ is

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is A $3$

$f\left(x\right)=6|x-1|+|3x-2|+2x$
$=\{\begin{array}{c}-6(x-1)-(3x-2)+2x,x<\frac{2}{3}\\ -6(x-1)+(3x-2)+2x,\frac{2}{3}\le x<1\\ 6(x-1)+(3x-2)+2x,x\ge 1\end{array}$

$=\{\begin{array}{c}(-\infty ,\frac{10}{3}),x<\frac{2}{3}\\ (3,\frac{10}{3}],\frac{2}{3}\le x<1\\ [3,\infty ),x\ge 1\end{array}$
Hence Range of f is $[3,\infty )$