Let $V \sim Uniform(0, a)$, and let $c_f, c_r$ be constants.
I need help finding the value of $T$ that minimizes the fraction
$$\frac{c_fP(V\leq T) + c_r P(V > T)}{\int_{0}^{T} P(V > x) \mathop{dx}} $$
I computed:
$$P(V\leq T) = \int_0^T \frac{1}{a} \mathop{dz} = T/a. $$
Also $P(V > T) = 1 - T/a$.
Then, I got
$$\int_0^T P(V > x) dx = T - T^2/2a. $$
Plugging it in, I get:
$$\frac{c_f (T/a)+c_r(1 - T/a)}{(T - T^2/2a)},$$
which I am supposed to minimize, if my other calculations are right. I need to find the value of $T$ that minimizes this. But I got $T \to \infty$ when I tried it. But I think this is wrong. I think it should be finite.
T > 0 by the way. It is time.
https://www.wolframalpha.com/input/?i=critical+points+%28bx%2Fa+%2B+c%281+-+x%2Fa%29%29%2F%28x+-+x%5E2%2F%282*a%29%29