Question

Weight of a person can be expressed as a function of their age. This function usually varies from person to person. Suppose this function is identical for two brothers and is monotonically increasing until the age of 50 years, after which it monotonically decreases. Let \( a_1 \) and \( a_2 \) (in years) denote the ages of the brothers, where \( a_1 < a_2 \).

Which one of the following statements is correct about their ages on the day they attain the same weight?

• A. \( a_1<a_2<50 \)
• B. \( a_1<50<a_2 \)
• C. \( 50<a_1<a_2 \)
• D. Either \( a_1 = 50 \) or \( a_2 = 50 \)

Hint:

In problems involving monotonic functions, focus on the turning point (in this case, age 50) to determine the behavior of the function for different values of the variables.

Answer 1

The Correct Option is B

The weight function is increasing until the age of 50 and then decreases. Given that \( a_1 < a_2 \), this means that the younger brother's age \( a_1 \) is less than 50 and the older brother's age \( a_2 \) is greater than 50. When both brothers reach the same weight, their ages must satisfy the condition where the younger brother is below 50 and the older brother is above 50. Hence, \( a_1 < 50 < a_2 \).