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

Question

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

Accepted Answer

To solve this problem:

1. **Understand the Components of $$ f(x) $$:**  
   - The term $$ |x - a| $$ represents the distance between $$ x $$ and $$ a $$.  
   - Similarly, $$ |x - 100| $$ is the distance between $$ x $$ and 100, and $$ |x - (a + 50)| $$ is the distance between $$ x $$ and $$ a + 50 $$.

2. **Break Down the Problem:**  
   Since $$ 0 \leq a, x \leq 100 $$, analyze how each term contributes to $$ f(x) $$.  
   The value of $$ f(x) = |x - a| + |x - 100| + |x - (a + 50)| $$ depends on the position of $$ x $$:  
   - $$ |x - a| + |x - (a + 50)| = (a + 50) - a = 50, 	ext{ when } x = a. $$

3. **Key Observations:**  
   - The fixed part of $$ f(x) = (100 - a) + (variable part that ranges from 0 to 50). $$
   - To maximize $$ f(x) = (100 - a) + 50, 	ext{ set } (100 - a) = 50.$$

4. Solve for a:   
   Setting the above condition gives:  
   $$
   a = 50.
   $$

Thus, the maximum value of $$ f(x) = 100 $$, and it occurs when **$$ a = 50 $$**.

Answer

Answer

Answer