How to approach the following optimization problem: $$\text{minimize }\int_{0}^{1}f(x)dx$$ over all (integrable) real-valued functions $f$ on $[0,1]$ satisfying $$1-x_{1}x_{2} \leq f(x_{1})f(x_{2})\text{ for all }x_{1},x_{2} \in [0,1]?$$

How to approach the following optimization problem: $$\text{minimize }\int_0^1 f(x) \, dx$$ over all (integrable) real-valued functions $f$ on $[0,1]$ satisfying $$1-x_1 x_2 \leq f(x_1)f(x_2)\text{ for all }x_1,x_2 \in [0,1]?$$