Question:medium

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
  • 100
  • 25
  • 0
  • 50
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The Correct Option is A

Solution and Explanation

Given the function:

\[ f(x) = |x - a| + |x - 100| + |x - (a + 50)| \] We aim to determine the value of \( a \) that maximizes \( f(x) \).

Step 1: Analyzing cases based on \( x \)

We consider three scenarios for \( x \) relative to \( a \) and \( a + 50 \):

  • Case 1: \( x \leq a \): \( f(x) \) is maximized at \( x = 0 \).
  • Case 2: \( a \leq x \leq a + 50 \): \( f(x) \) is maximized at \( x = a \).
  • Case 3: \( x \geq a + 50 \): \( f(x) \) is maximized at \( x = 100 \).

Step 2: Comparing maximum values across cases

We compare the maximum values derived from each case: - In Case 1, the maximum value is \( 2a + 150 \). - In Case 2, \( f(x) \) reaches its maximum at \( x = a \). - In Case 3, the maximum value occurs at \( x = 100 \).

Step 3: Maximizing \( f(x) \)

To maximize \( f(x) \), we must maximize the expression \( 2a + 150 \), which occurs when \( a = 100 \).

Final Answer:

The maximum value of \( f(x) \) is 100, attained when \( a = 100 \).

Conclusion:

The correct answer is \( \boxed{100} \).

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