Let's see if I got it.
Suppose I want to find the best constant for the inequality
\int_0^\mu f(x) dx \le K \int_0^1 f(x) dx,
where f(x) \in C^1(0,1), f(0) = 0, f(x) \ge 0, and 0 \le \mu \le 1.
Let
f_n(x) = \begin{cases} \frac{n+2}{n+3} x(2\mu -x), &0 < x \le \mu, \\
\\
\frac{n+2}{n+3}...