12.6.4 Higher mathematics

Problem: Find all such \( f \in C^{2}[0 ; 1] \) functions that 1) \( f(0)=f^{\prime}(0)=1 \), 2) \( f^{\prime \prime}(x) \geq 0, \forall x \in(0 ; 1) \), 3) \( \int_{0}^{1} f(x) d x=\frac{3}{2} \).