Question

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

• A. 100
• B. 25
• C. 0
• D. 50

Answer 1

The Correct Option is A

We are given the function:

\[ f(x) = |x - a| + |x - 100| + |x - (a + 50)| \] and asked to find the value of \( a \) that maximizes \( f(x) \).

Step 1: Analyzing the cases based on the value of \( x \)

We analyze three cases based on the value of \( x \) relative to \( a \) and \( a + 50 \):

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

Step 2: Comparing the maximum values in each case

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

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

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

Final Answer:

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

Conclusion:

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