Let $1\le p \le \infty$. Prove that for all $\epsilon >0$ there exists a constant $C>0$ such that $$\|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \; \forall u\in W^{2,p}(\mathbb{R}).$$ This is a special case of an interpolation inequality, see here http://conteudo.icmc.usp.br/pessoas/andcarva/sobolew.pdf , theorem 2.
I need help to prove this. In any case the first step is to choose a continuous representative function for u. Then, I have the following 2 ideas:
1) To do it similarly as to prove theorem 2. Apply the mean value theorem to u to receive an estimation for $|u'|$. But the first problem I have here is, that the mean value theorem is local. But here the domain is $\mathbb{R}$. Therefore, i don't know how to proceed.
2) To use the fundamental theorem of anlysis applied to $u'$, to receive something like $$u'(\delta+x)=u'(x)+\int_x^{x+\delta} u''(z)dz.$$ Furthermore, convexity of $|\cdot |^p$ and Jensens inequality https://de.wikipedia.org/wiki/Jensensche_Ungleichung might be important, but I am not completely sure how to math that.
The second idea may be more promising. I appreciant any help.
Edit: There are other variants of this inequality available and meanwhile I have discovered Interpolation inequality and Proving an interpolation inequality for $C^2_b$ functions which seem to be a variant of the inequality above (however, the norms are different).
In fact, I think you only need to prove for $u\in C_c^{\infty}(\mathbb{R})$ which is a dense subset of $W^{1,2}(\mathbb{R})$ (But NOT dense in $W^{1,2}[0,100]$). Then the desired result holds by pushing limit. I hope I remembered it right. :)