Question Statement:
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 a1 and a^2 (in years) denote the ages of the brothers and a1 < a^2.
Which one of the following statements is correct about their age on the day when
they attain the same weight?
Options:
-
(A) a1 < a2 < 50
-
(B) a1 < 50 < a2
-
(C) 50 < a1 < a2
-
(D) Either a1 = 50 or a2 = 50
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.7)**
**1. Intuitive Physical Setup & Given Parameters**:
- Person's weight $W(a)$ is an identical function of age $a$ for two brothers.
- $W(a)$ is **strictly monotonically increasing** for $a < 50$.
- $W(a)$ is **strictly monotonically decreasing** for $a > 50$.
- Peak weight occurs at age $a = 50$.
- Let $a_1$ and $a_2$ be the ages of the two brothers with $a_1 < a_2$.
- On a certain day, they have the **same weight**: $W(a_1) = W(a_2)$.
- We need to determine the correct statement regarding their ages.
**2. Step-by-Step Mathematical Derivation**:
1. **Properties of Strictly Monotonic Functions**:
- If two distinct inputs produce the same output on a function ($W(a_1) = W(a_2)$ with $a_1 \neq a_2$), the function **cannot be strictly monotonic** on any interval containing both inputs.
- For $a \in [0, 50]$, $W(a)$ is strictly increasing. Hence, if $a_1 < a_2 \le 50$, then $W(a_1) < W(a_2)$, which contradicts $W(a_1) = W(a_2)$. Therefore, both ages cannot be $\le 50$.
- For $a \in [50, \infty)$, $W(a)$ is strictly decreasing. Hence, if $50 \le a_1 < a_2$, then $W(a_1) > W(a_2)$, which also contradicts $W(a_1) = W(a_2)$. Therefore, both ages cannot be $\ge 50$.
2. **Intersection Across the Peak**:
- For $W(a_1) = W(a_2)$ to hold with $a_1 < a_2$, one brother must be on the ascending side of the curve and the other brother must be on the descending side of the curve:
a_1 < 50 < a_2
**3. Comprehensive Options & Distractor Analysis**:
- **(A) a1 < a2 < 50**: ❌ **INCORRECT**. Violates strict monotonicity on $[0, 50)$ (would imply $W(a_1) < W(a_2)$).
- **(B) a1 < 50 < a2**: ✅ **CORRECT**. One brother is younger than 50 (ascending branch) and the other is older than 50 (descending branch).
- **(C) 50 < a1 < a2**: ❌ **INCORRECT**. Violates strict monotonicity on $(50, \infty)$ (would imply $W(a_1) > W(a_2)$).
- **(D) Either a1 = 50 or a2 = 50**: ❌ **INCORRECT**. The peak value $W(50)$ is unique and cannot equal any other value since $W(a) < W(50)$ for all $a \neq 50$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: By Rolle's theorem / injectivity of monotonic functions, identical values of a unimodal function must lie on opposite sides of the global extremum.