Question

The minimum value of f(x) = $|x-1|$ +$|x-2|$ + $|x-3|$

• A. 1
• B. 2
• C. 3
• D. 0

Answer 1

The correct option is B

2

To find the minimum value, we will first remove the modulus.

If x $\le$ 1,$|x-1|$ = 1 - x,$|x-2|$ = 2 - x and |$|x-3|$ = 3 - x

Similarly, we can split f(x) into the following intervals.

If x $\le$ 1, f(x) = 1 - x + 2 - x + 3 - x

= 6 - 3x

Similarly,

f(x) = 3x - 6 if x $\ge$ 3

= x if 2 $\le$ x $\le$ 3

= 4 - x if 1 $\le$ x $\le$ 2

= 6 - 3x if x $\le$ 1

We will find the minimum value in each interval and then find the minimum of those values.

if x $\ge$ 3 f(x) = 3x - 6. The minimum value occurs at x = 3, f(3) = 3

If 2 $\le$ x $\le$ 3, f(x) = x. Minimum value occurs at x = 2 f(2) = 2
If 1 $\le$ x $\le$ 2, f(x) = 4-x. Minimum value occurs at x=2 beacuse higher the value of x lesser is the value of the function. So minimum value is 4-2 = 2
If x $\le$ 1 , f(x) = 6-3x = minimum value will occur at x = 1 beacuse higher the value of x lesser is the value of the function. So minimum value = 6-3 = 3.

Out of all these minimum values the minimum value = 2