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If $f:[0,1]\to\mathbb{R}$$f:[0,1]^n\to\mathbb{R}$ is $L$-Lipschitz and $f(0)=0$, then $|f(x)|\le L\|x\|$, which obviously implies that the function $x\mapsto L\|x\|$ is the maximizer for both problems.
If $f:[0,1]\to\mathbb{R}$ is $L$-Lipschitz and $f(0)=0$, then $|f(x)|\le L\|x\|$, which obviously implies that the function $x\mapsto L\|x\|$ is the maximizer for both problems.
If $f:[0,1]^n\to\mathbb{R}$ is $L$-Lipschitz and $f(0)=0$, then $|f(x)|\le L\|x\|$, which obviously implies that the function $x\mapsto L\|x\|$ is the maximizer for both problems.
If $f:[0,1]\to\mathbb{R}$ is $L$-Lipschitz and $f(0)=0$, then $|f(x)|\le L\|x\|$, which obviously implies that the function $x\mapsto L\|x\|$ is the maximizer for both problems.
