CAT 2022 Slot 1 QA Question & Solution
AlgebraMedium
Question
Let $0 \leq a \leq x \leq 100$ and $f(x) = \mid x - a \mid + \mid x - 100 \mid + \mid x - a - 50\mid$. Then the maximum value of f(x) becomes 100 when a is equal to
Options
25
100
50
0
Solution
When $x \ge a$, we have:
$$
|x - a| = x - a
$$
Also, when $x < 100$, we have:
$$
|x - 100| = 100 - x
$$
Now, the function $f(x)$ is given by:
$$
f(x) = (x - a) + (100 - x) + |x - a - 50| = 100
$$
Simplifying the equation:
$$
|x - a - 50| = a
$$
From the graph, we can see that when $x = a$, we get:
$$
|x - a - 50| = a
$$
This simplifies to:
$$
a = 50
$$
Similarly, when $x = a + 100$, we also get:
$$
|x - a - 50| = a
$$
This gives:
$$
a = 50
$$
Thus, the value of $a$ is 50 when $f(x) = 100$.