I am a bit confused as to what role the Chebyshev nodes play in the optimization of Langrage Polynomial Interpolation.
Reading online for what I understand that the Chebyshev nodes gives us the minimum error, I may be doing something wrong but take a look at the following example:
Approximating $f(x) = x^3$ with a linear Langrage interpolating polynomial with the Chebyshev nodes gives us that $P_1(x) = 0.5x$, this gives us a maximum error of 0.5 on the interval $[-1,1]$.
but if I had chosen the nodes $x_1=\sqrt0.6 , x_2= -\sqrt0.6$, then my interpolating polynomial becomes $P_1^*(x) = 0.6x$ giving us a maximum error of 0.4 on the same interval.
I know I must be misunderstanding something. What gives?