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Let \(0≤a≤x≤100\) and f(x)=\(|x−a|+|x−100|+|x−a−50|\).Then the maximum value of f(x) becomes 100 when a is equal to

Updated On: Jan 15, 2026
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Solution and Explanation

The given function is: \(f(x) = |x - a| + |x - 100| + |x - a - 50|\). This function calculates the sum of the distances from a point \(x\) to three fixed positions: \(a\), 100, and \(a + 50\).

The function can be rewritten for clarity as: \(f(x) = |x - a| + |x - 100| + |x - (a + 50)|\)

Interpretation of Terms:

  • \(|x - a|\): Represents the distance between \(x\) and \(a\).
  • \(|x - 100|\): Represents the distance between \(x\) and 100.
  • \(|x - (a + 50)|\): Represents the distance between \(x\) and \(a + 50\).

Given the condition \(a \leq x \leq 100\), it follows that the sum \(|x - a| + |x - 100|\) simplifies to \(100 - a\). This value is a constant and does not depend on \(x\).

Therefore, the function can be simplified to: \(f(x) = (100 - a) + |x - (a + 50)|\)

Analysis of \( |x - (a + 50)| \)

To determine the maximum value of \(f(x)\), we need to maximize the term \(|x - (a + 50)|\). Two scenarios are considered:

Scenario 1: \( x \) is between \( a \) and \( a + 50 \)

In this case, the maximum distance from \(x\) to \(a + 50\) occurs when \(x = a\). The distance is \(|a - (a + 50)| = 50\). Consequently, the maximum function value is \(f(x) = (100 - a) + 50\).

Scenario 2: \( x \geq a + 50 \)

To maximize \(|x - (a + 50)|\), we consider the extreme values within the given constraints. Setting \(a = 0\) and \(x = 100\), we have \(a + 50 = 50\). The distance becomes \(|100 - 50| = 50\). The function value is then \(f(x) = (100 - 0) + 50 = 150\).

Conclusion:

In both analyzed scenarios, the maximum possible value for \(|x - (a + 50)|\) is 50. This is the key to finding the maximum value of \(f(x)\).

Significance of Maximizing \( |x - (a + 50)| \)
Since the term \(|x - a| + |x - 100| = 100 - a\) is constant with respect to \(x\), the maximization of \(f(x)\) depends solely on the maximization of the variable term \(|x - (a + 50)|\). Determining its maximum value directly informs the overall maximum value of \(f(x)\).

By maximizing \(|x - (a + 50)|\), we ensure \(f(x)\) is maximized. The highest value \(f(x)\) can attain is \(100 - a + 50 = 150\), which occurs when \(a = 0\) and \(x = 100\).

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