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Suppose that $f \in C^\infty(\mathbb R)$ has compact support in $[-T,T]$ where $T>0$. Is it possible that the $L^1$-norm of its derivatives are growing exponentially? That means $$ \| f^{(n)} \|_1 \to \infty $$ exponentially. For me it's hard to imagine that such a function could exist. Is there a way to construct such a function or argue that it cannot exist?

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Take $e^{e^x}$ on $[0,1]$ and smooth it down to zero at the boundary of your interval (this will only increase the $L^1$-norm). Note $f^{(n)}(x)$ indeed grows exponentially and thus so does its $L^{1}$-norm.

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