CAT 2006 — QA Question 16
Answer the next 2 questions based on the information given below:
A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.
Let f(x) = max (2x + 1, 3 − 4x), where x is any real number. Then the minimum possible value of f(x) is:
Answer & solution
- A
1/3
- B
1/2
- C
2/3
- D
4/3
5/3
Let the two lines represent the equations y = 2x + 1 and y = 3 – 4x
f(x) = max{2x + 1, 4 - 3x} is represented by green and blue lines.
f(x) is greater than 5/3, when x < 1/3 (green part) or x > 1/3 (blue part).
∴ f(x) is minimum at x = 1/3 and this value is 5/3.
Hence, option (e).
Note: The minimum value of the function f(x) = max (ax + b, cx + d) occurs when ax + b = cx + d (Only if one line has a positive slope and the other one has a negative slope)