Let $x_0=t$ and $x_1=1-t$. Find $t\in[0,1]$ such that $\max_{x\in[0,1]}|w_1(x)|$ is minimal. Whereas $w_1(x) = (x-x_0)(x-x_1)$. I don't know how to find a specific value for t here. Anyway here's my attempt: I found that
$|w_1(x)| = w(x)$ if $x\in[0,t]$ or $x\in[1-t,1]$ and
$|w_1(x)| = -w(x)$ if $x\in [t,1-t]$
I now need to find $\min_{t\in[0,1]}\max_{x\in[t,1-t]}(-w(x))$. The expression $\max_{x\in[t,1-t]}(-w(x))$ is maximal at $x=1/2$ (I figured that from a plot I made of $|w(x)|$). So I think I have to find $\min_{t\in[0,1]}(-t^2+t+1/4)$. Am I going anywhere with this?