Consider approximating the function $f(x)=x^{3}-x$ by a polynomial $P_{2}(x)=a_{2} x^{2}+$ $a_{1} x+a_{0}$ which minimizes $$ E_{2}\left(a_{0}, a_{1}, a_{2}\right)=\int_{0}^{1}\left[f(x)-P_{2}(x)\right]^{2} d x $$ Find a polynomial $P_{1}$ that solves $$ \min _{P_{1} \in \Pi_{1}} \max _{x \in[0,1]}\left|P_{2}(x)-P_{1}(x)\right| $$ where $\Pi_{1}$ denotes a set of all polynomials degree at most $1 .$
It seems that such similar problems are mostly related to Chebyschev's polynomials, but I could relate precisely how these exercises could potentially yield to the inequalities involved with Chebyschev's ones? What are the major instincts or ideas behind those problems?