Here is an example to better explain my question.
Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$.
First approach - Taylor polynomials
$$f(x) \simeq g(x) = 1 - \frac{x^2}{2} $$
Second approach - Minimization of Euclidean Norm
I look for $g(x)$ such that:
$$[a, b, c] = \arg \min_{a,b,c} \Phi(a,b,c),$$
where
$$\Phi(a,b,c) = \int_{-\pi}^{+\pi}[f(x)-g(x)]^2\text{d}x$$ I have to evaluate first derivative with respect to $a$, $b$ and $c$ and impose that they are $0$:
$$\frac{\partial \Phi}{\partial a} = \int_{-\pi}^{+\pi}\frac{\partial }{\partial a}[f(x)-g(x)]^2\text{d}x = -2\int_{-\pi}^{+\pi}[f(x)-g(x)]\text{d}x = 0$$ $$\frac{\partial \Phi}{\partial b} = \int_{-\pi}^{+\pi}\frac{\partial }{\partial b}[f(x)-g(x)]^2\text{d}x = -2\int_{-\pi}^{+\pi}x[f(x)-g(x)]\text{d}x = 0$$ $$\frac{\partial \Phi}{\partial c} = \int_{-\pi}^{+\pi}\frac{\partial }{\partial c}[f(x)-g(x)]^2\text{d}x = -2\int_{-\pi}^{+\pi}x^2[f(x)-g(x)]\text{d}x = 0$$
At the end, I get:
$$g(x) = \frac{15}{2\pi^2} - \frac{45x^2}{2\pi^4}$$
Other approaches
One can use a different norm
The question
Are the Taylor polynomials of a function the results of a minimization problem?
I'm not sure how satisfying this answer will be, but here's something:
The $n$th degree Taylor (MacLaurin) polynomial $g(x)$ of a function $f(x)$ is the unique degree $n$ polynomial such that $$ \lim_{x \to 0} \left(\frac{f(x) - g(x)}{x^n}\right)^2 = 0 $$ Is there some corresponding quantity that has been minimized? I'm not sure