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 $C = 1 + \frac{1}{x_2 - x_1}$.