My opimization problem is related to finding an optimal probability density function (pdf) $f(t)$ to minimize the following objective.
Find $\min\limits_{f(t)} C$ such that:$$1+\frac{b\cdot\int_{0}^{x}f(t)dt}{x}\leq C, \text{ for any }x\in(0,\infty)$$ $$\frac{\int_{0}^{x}(t+b)\cdot f(t)dt+x\cdot\int_{x}^{\infty}f(t)dt}{b}\leq C, \text{ for any }x\in(b,\infty)$$ $$\int_{0}^{\infty}f(t)dt=1$$ Here $b$ is a known constant.
I want to find the optimal pdf to minimize $C$, but do not know how to solve it.
Are there any systematic ways or general ideas of dealing with such a functional optimization problem?