Interpolation Inequality in finite dimensional linear space

Question

How can we prove -

$$|\dot{u}(x) - \tilde{u_h}(x)| \leqslant h \left[\max_{0 \leqslant y\leqslant1}\ddot{u}(y)\right],$$

where $0\leqslant x\leqslant 1$ and $\tilde{u_h}$ is interpolant of $u$.