Question:
The function \( f(x) = |x - 2| + |2.5 - x| + |3.6 - x| \), where \( x \) is a real number, attains a minimum at
The function \( f(x) = |x - 2| + |2.5 - x| + |3.6 - x| \), where \( x \) is a real number, attains a minimum at
Show Hint
The function adds the distances of \( x \) from three fixed points. Think about which of the three points a walker should stand at to keep the total distance smallest.
Updated On: Nov 7, 2026
- \( x = 2.3 \)
- \( x = 2.5 \)
- \( x = 2.7 \)
- None of the above
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
The function \(f(x) = |x - 2| + |2.5 - x| + |3.6 - x|\) adds the distances of \(x\) from three points on the number line: \(2\), \(2.5\) and \(3.6\). We need the value of \(x\) where this total distance is the least.
The graph of such a sum is made of straight pieces. The slope of each piece changes only at the three points \(2\), \(2.5\) and \(3.6\).
Step 2: Slope on each stretch:
Each term \(|x - c|\) has slope \(+1\) when \(x\) is to the right of \(c\), and slope \(-1\) when \(x\) is to the left of \(c\). We add the three slopes.
For \(x < 2\): all three points are to the right, so the slope is \(-1 - 1 - 1 = -3\). The function is falling.
For \(2 < x < 2.5\): the point \(2\) is on the left (slope \(+1\)), the other two are on the right (slope \(-1\) each). Total slope \(= 1 - 1 - 1 = -1\). Still falling.
For \(2.5 < x < 3.6\): two points are on the left and one on the right. Total slope \(= 1 + 1 - 1 = +1\). Now the function is rising.
For \(x > 3.6\): total slope \(= +3\). Rising even faster.
Step 3: Locate the minimum:
The function falls until \(x = 2.5\) and rises after it, so the minimum is at \(x = 2.5\), the middle one of the three points. The least value is
\[ f(2.5) = 0.5 + 0 + 1.1 = 1.6 \]
Step 4: Check the given options:
Option 1, \(x = 2.3\): \(f = 0.3 + 0.2 + 1.3 = 1.8\), which is more than \(1.6\). So option 1 is wrong.
Option 2, \(x = 2.5\): \(f = 1.6\), the least value. So option 2 is correct.
Option 3, \(x = 2.7\): \(f = 0.7 + 0.2 + 0.9 = 1.8\), again more than \(1.6\). So option 3 is wrong.
Option 4 is wrong, because a listed value already gives the minimum.
Final Answer:
The sum of distances is smallest at the middle point, \(x = 2.5\), which is option 2. \[ \boxed{x = 2.5 \text{ (Option 2)}} \]
The function \(f(x) = |x - 2| + |2.5 - x| + |3.6 - x|\) adds the distances of \(x\) from three points on the number line: \(2\), \(2.5\) and \(3.6\). We need the value of \(x\) where this total distance is the least.
The graph of such a sum is made of straight pieces. The slope of each piece changes only at the three points \(2\), \(2.5\) and \(3.6\).
Step 2: Slope on each stretch:
Each term \(|x - c|\) has slope \(+1\) when \(x\) is to the right of \(c\), and slope \(-1\) when \(x\) is to the left of \(c\). We add the three slopes.
For \(x < 2\): all three points are to the right, so the slope is \(-1 - 1 - 1 = -3\). The function is falling.
For \(2 < x < 2.5\): the point \(2\) is on the left (slope \(+1\)), the other two are on the right (slope \(-1\) each). Total slope \(= 1 - 1 - 1 = -1\). Still falling.
For \(2.5 < x < 3.6\): two points are on the left and one on the right. Total slope \(= 1 + 1 - 1 = +1\). Now the function is rising.
For \(x > 3.6\): total slope \(= +3\). Rising even faster.
Step 3: Locate the minimum:
The function falls until \(x = 2.5\) and rises after it, so the minimum is at \(x = 2.5\), the middle one of the three points. The least value is
\[ f(2.5) = 0.5 + 0 + 1.1 = 1.6 \]
Step 4: Check the given options:
Option 1, \(x = 2.3\): \(f = 0.3 + 0.2 + 1.3 = 1.8\), which is more than \(1.6\). So option 1 is wrong.
Option 2, \(x = 2.5\): \(f = 1.6\), the least value. So option 2 is correct.
Option 3, \(x = 2.7\): \(f = 0.7 + 0.2 + 0.9 = 1.8\), again more than \(1.6\). So option 3 is wrong.
Option 4 is wrong, because a listed value already gives the minimum.
Final Answer:
The sum of distances is smallest at the middle point, \(x = 2.5\), which is option 2. \[ \boxed{x = 2.5 \text{ (Option 2)}} \]
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