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 \(a_1 \lt 50 \lt a_2\).
Both brothers follow the same weight function w, which is strictly increasing up to 50 and strictly decreasing after. If both ages were below 50, \(a_1 \lt a_2\) would give \(w(a_1) \lt w(a_2)\); if both were above 50, it would give \(w(a_1) \gt w(a_2)\). Either way the weights differ.
Equal weights at different ages are only possible when the ages lie on opposite sides of the peak, one rising and one falling. If one of them were exactly 50 he would be at the unique maximum, which the other, at a different age, cannot match. So the younger brother is under 50 and the elder over 50.
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