Do we have $\|f\|_\infty\leq |f(0)|+\|f'\|_\infty$? Do we have that for any $C^1([0,1],\mathbb{R})$ function $f$ this: $\|f\|_\infty\leq |f(0)|+\|f'\|_\infty$?

Let $x\in [0,1]$, the MVT theorem implies that $f(x)-f(0)=f'(c_x)x, c_x\in (0,x)$ this implies that $|f(x)|\leq |f(0)|+|f'(c_x)|$ and $\|f\|_{\infty}\leq |f(0)|+\|f'\|_{\infty}$.
