# Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following:

$$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$

and

$$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$

Note that $e^{-x}(e^xf(x))'=f(x)+f'(x)$. So $\int_0^1(|f(t)|+|f'(t)|)dt\ge|\int_0^1(f(t)+f'(t))dt|\ge\int_0^1(e^tf(t))'dt=ef(1)-f(0)$

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