Discussion
Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let \(a_{1}\) and \(a_{2}\) (in years) denote the ages of the brothers and \(a_{1} \lt a_{2}\).
Which one of the following statements is correct about their age on the day when they attain the same weight?
Answer: B
The correct statement is \(a_{1} \lt 50 \lt a_{2}\).
The weight function rises strictly up to 50 years and falls strictly after it. On the rising part, different ages give different weights, and the same holds on the falling part. So two different ages can give the same weight only if one is on the rising side and the other on the falling side: the younger brother is under 50 and the elder over 50.
If both ages were below 50, or both above 50, strict monotonicity would make their weights different. Neither age can be exactly 50, since the weight at 50 is the maximum and is reached at no other age. That rules out the other three options.
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