I would like to find the best linear approximation for the function ($x\in \mathbb{R}$) $$f(x)=x^{-4}$$ in the range $$x\in [x_1, x_2]$$ The motivation for this is that exponential functions, as well as logarithmic functions and others, all asymptote to approach $0$ when $x$ is large, the change of slope at $x$ values away from $x=0$ is also not significant. Therefore the behavior of the function at large $x$ should be well approximated by a linear function.
My idea is to find $\alpha, \beta \in \mathbb{R}$ for $$g(x)=\alpha x+\beta$$ such that $$\int_{x_1}^{x_2} [f(x)-g(x)]^2 \,dx$$ is minimized.
Is my thought correct? How could I continue?