I wish to find $C>0$ such that $$|f(1/3)| \leq C\|f\|_{W^{1,2}[0,1]},$$ for all $f$ in the Sobolev space $W^{1,2}[0,1]$.
The context is that $\rho(f)=|f(1/3)|$ defines a norm on the space of zero degree polynomials on the unit interval. If we can find such constant above then by a theorem we may conclude $$\rho’(f):=\left(|f(1/3)|^2 + \int_0^1 |f(x)|^2 dx \right)^{1/2}$$ is a norm on $W^{1,2}$ equivalent to the standard Sobolev norm. But I am absolutely stumped on finding this constant. I’ve tried to do a proof by contradiction and modeled after a similar inequality involving Sobolev/$L^p$ norms, but I cannot pinpoint the exact argument.
This is for homework, so please hints only.