CAT 2022 Slot 1 QA Question 6: 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

Since $0 \leq a \leq x \leq 100$, the first two modulus terms can be simplified directly.
$|x-a| = x-a$
$|x-100| = 100-x$
$f(x) = 100-a + |x-a-50|$
Let $t = x - a$. Then $t$ ranges from $0$ to $100 - a$.
$f(x) = 100-a + |t-50|$
The maximum possible value of $|t - 50|$ on this interval is $50$, attai

Let 0≤a≤x≤1000 \leq a \leq x \leq 1000≤a≤x≤100 and f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣f(x) = |x - a| + |x - 100| + |x - a - 50|f(x)=∣x−a∣+∣x−100∣+∣x−a−50∣. Then the maximum value of f(x)f(x)f(x) becomes 100100100 when aaa is equal to

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