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're given the function \(f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣\) and asked to find the value of a that maximizes f(x). We analyze three
cases for the possible values of x relative to a and a+50.
1. \(x≤a\): Maximized at x=0.
2. \(a≤x≤a+50\): Maximized at x=a.
3. \(x≥a+50\): Maximized at x=100.
Comparing the maximum values in each case, we find that the maximum value of f(x) occurs in Case 1, where 2a+150 is the expression for the
maximum value.
To maximize f(x), we need to maximize 2a+150, which is achieved when a=100.
So, the maximum value of f(x) is 100, and it happens when a is equal to 100. Thus, the correct answer is: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 \(a = 50. \)