Best minimum constant for a functional inequality

Question

Let $f$ be a twice differentiable function from $[0,1]$ to $\mathbb{R}$ with $f"$ continuous on $[0,1]$ and $\int_a^b f(x)dx=0$ where $0

$$\frac{\left(\int_0^1 f(x) dx\right)^2}{\int_0^1 (f^{\prime\prime}(x))^2 dx} \le C$$

I tried to write $\int_0^a f^{\prime\prime}(x) P_1(x) dx$, $\int_0^a f^{\prime\prime}(x) P_2(x) dx$ and $\int_0^a f^{\prime\prime}(x) P_3(x) dx$ where $P_1$, $P_2$ and $P_3$ are three polynomials, as function of $\int_0^a f(x) P_1(x^{\prime\prime}) dx$, $\int_0^a f(x) P_2^{\prime\prime}(x) dx$ and $\int_0^a f(x) P_3^{\prime\prime}(x) dx$ using integration by part and tried to apply Cauchy-Schwartz inequality but without success.