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)
Explanation
Let \(w\) be the common weight function, taken as strictly increasing up to age 50 and strictly decreasing after it, so its single peak is at 50.
- If both ages are at most 50, then \(a_{1} \lt a_{2}\) gives \(w(a_{1}) \lt w(a_{2})\).
- If both ages are at least 50, then \(w(a_{1}) \gt w(a_{2})\).
Either way the weights differ. These two cases cover options A and C, and also D, because an age of exactly 50 falls into one of them. Equal weights are therefore possible only with the ages on opposite sides of the peak: \(a_{1} \lt 50 \lt a_{2}\).
Answer: B