I'm trying to prove the following
Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function supported on $[a, b]$ where $-\infty < a < b < \infty$. $2 \le p < \infty$. Then $$ \|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2}. $$
However, I got stuck since I can't find a way to effectively relate the three (I can relate $f$ and $f'$ or $f'$ and $f''$ by the Fundamental Theorem of Calculus but the one that is left out is a headache). Could anyone give me a hint on this? Thanks.
If $I$ is an interval where $f'$ is positive, we have, by integration by parts, $$\int_I |f'(x)|^pdx=\int_If'(x)f'(x)^{p-1}dx=-(p-1)\int_If(x)f''(x)f'(x) ^{p-2}dx,$$ and similarly if $f'$ is non-negative. By summing over intervals $I$ where $f'$ has constant sign, we obtain $$\Vert f'\Vert_{L^p}^p=\int_a^b|f'(x)|^pdx\leq (p-1)\int_a^b|f(x)||f''(x)||f'(x)|^{p-2}dx.$$ By Hölder's inequality, since $(1/p)+(1/p)+(p-2)/p=1$, we obtain $\int|f||f''|f'|^{p-2}\leq\Vert f\Vert_{L^p}\Vert f''\Vert_{L^p}\Vert|f'|^{p-2}\Vert_{L^{p/(p-2)}}=\Vert f\Vert_{L^p}\Vert f''\Vert_{L^p}\Vert f'\Vert_{L^p}^{p-2}$, thus $$\Vert f'\Vert_{L^p}^2=\Vert f'\Vert_{L^p}^{p-(p-2)}\leq(p-1)\Vert f\Vert_{L^p}\Vert f''\Vert_{L^p}.$$