Mathematics

Problem

Let

n \ge 2

be a natural number and

f: [0,1] \to \mathbb{R}

be the function defined by

f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \le x \le \frac{1}{2n} \\ 2n(2nx - 1) & \text{if } \frac{1}{2n} \le x \le \frac{3}{4n} \\ 4n(1 - nx) & \text{if } \frac{3}{4n} \le x \le \frac{1}{n} \\ \frac{n}{n-1}(nx - 1) & \text{if } \frac{1}{n} \le x \le 1 \end{cases}

n

is such that the area of the region bounded by the curves

x = 0

x = 1

y = 0

and

y = f(x)

is 4, then the maximum value of the function

f

is [JEE-Advanced 2023]