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

Answer 1

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

Let us rewrite the function to group the terms more clearly: \(f(x) = |x - a| + |x - 100| + |x - (a + 50)|\)

Understanding Each Term:

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

We are told that: \(a \leq x \leq 100\).
From this, note that: \(|x - a| + |x - 100|\) = \(100 - a\), which is constant and independent of x.

So, we can simplify the function as: \(f(x) = (100 - a) + |x - (a + 50)|\)

Analyzing \( |x - (a + 50)| \)

To understand how to maximize \(f(x)\), we must consider how to maximize \(|x - (a + 50)|\). There are two possibilities:

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

In this scenario, the farthest \( x \) can be from \( a + 50 \) is exactly 50 units, and this happens when \(x = a\). So: \(|x - (a + 50)| = |a - (a + 50)| = 50\)
Thus, the maximum value of the function is: \(f(x) = (100 - a) + 50\)

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

To maximize \(|x - (a + 50)|\) again, we take \(x\) to its rightmost possible value (i.e., 100), and \(a\) to its leftmost possible value (i.e., 0).

Then: \(a = 0 \Rightarrow a + 50 = 50\) and \(x = 100\)
So, \(|x - (a + 50)| = |100 - 50| = 50\)
Again, the function becomes: \(f(x) = (100 - 0) + 50 = 150\)

Conclusion:

In both cases, the maximum value that \(|x - (a + 50)|\) can take is 50, and that’s the key to finding the maximum value of the function.

Why is understanding the maximum of \(|x - (a + 50)|\) important?
Because the rest of the expression, \(|x - a| + |x - 100| = 100 - a\), is constant with respect to x, we focus on the variable part: \(|x - (a + 50)|\). Finding its maximum helps in determining the overall maximum value of \(f(x)\).

Hence, by understanding and maximizing \(|x - (a + 50)|\), we ensure \(f(x)\) is maximized. The maximum value is: \(f(x) = 100 - a + 50 = 150\), when \(a = 0\) and \(x = 100\)

Answer 2

Because \(x \geq a\), \(|x-a| = x-a \)
\((x-a) + (100-x) + |x-a-50| = 100 |x-a-50| = a = f(x)\)


The graph indicates that when \(x=a, a= 50\) and \(|x-a-50|=a.\)
Likewise, if \(x=a+100\), then \(|x-a-50|=a,\) and \(a= 50.\)
Thus, when\( f(x) = 100\), the value of \(a = 50. \)