Question

The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at

• A. x = 2.3
• B. x = 2.5
• C. x = 2.7
• D. None of the above

Answer 1

The correct option is B

x = 2.5

When a minimum question is asked for a modulus function we should always try to make one of the components zero ( as that is the minimum the modulus values can take). Hence we take x=2, x= 2.5 and x=3.6 and check where the value of y is minimum.

Case 1: If x $<$ 2, then y = 2 - x + 2.5 - x + 3.6 - x = 8.1 - 3x.
This will be least if x is highest i.e. just less than 2.
In this case y will be just more than 2.1

Case 2: If 2 $\le$ x$<$ 2.5 , then y = x - 2 + 2.5 - x 3.6 - x = 4.1 - x
Again, this will be least if x is the highest case y will be just more than 1.6.

Case 3: If 2.5$\le$ x$<$ 3.6 , then y = x - 2 + x - 2.5 + 3.6 - x = x - 0.9
This will be least if x is least i.e. X = 2.5.

Case 4: If In this case y = 1.6 X$\ge$ 3.6 , then
y = x - 2 + x - 2.5 + x - 3.6 = 3x - 8.1
y = x - 2 + x - 2.5 + x - 3.6 = 3x - 8.1
The minimum value of this will be at x = 3.6 = 27
Hence the minimum value of y is attained at x = 2.5

Shortcut:
Put x=2, f(x) = 2.1
When x=2.5, f(x) = 1.6
When x=3.6, f(x) = 2.7

Therefore, minimum value is attained at x=2.5.