We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Conclude the case $K = 2$ by showing the interpolation inequality
$$\forall f \in \mathcal{C}^{2}([0,1]), \quad \max\left(\|f\|_{\infty}, \left\|f^{\prime}\right\|_{\infty}\right) \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + C\left(\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|\right)$$
with $C = 1 + \frac{1}{x_{2} - x_{1}}$.