How can I find best approximation of a polynomial $f(x) = -2x^3+3x^2-4x+5$ with respect to max norm by a polynomial of degree 2 on segment [1,5].
As I have understood, if I have a polynomial $g(x) = ax^2+bx+c$, then I'd have to minimize $\begin{equation} \underset{1 \leq x \leq 5}{\max}|f(x)-g(x)| \end{equation}$ but I'm not exactly sure how to proceed further. Any help would be appreciated and thanks in advance!.
Hint
Consider that this is the same as a quadratic regression over an infinite number of data points.
So, the equivalent is to consider $$\Phi(a,b,c)=\int_1^5 \big(f(x)-g(x)\big)^2\, dx$$ Compute it and set the three equations $$\frac{\partial \Phi(a,b,c)}{\partial a}=\frac{\partial \Phi(a,b,c)}{\partial b}=\frac{\partial \Phi(a,b,c)}{\partial c}=0$$
You should have three linear equations for three variables.