Consider a function $g:\mathbb R^3\to\mathbb R$ defined by $$g(x,y,z)=(x+ay+bz+c)^2+(x+dy+ez+f)^2.$$ Here, $a$, $b$, $\cdots$, $f$ are real numbers. Does $g$ has minimum? (i.e. How can I ensure the existence of the minimum of the function?)
My trial : I originally thought that $g$ is a quadratic form and tried to check the positive definiteness. But it is not a quadratic form as it stand.
(The motivation for this problem is the linear regression. I wondered why $\text{SS}_{\text{reg}}=\sum_{i=1}^k\left(y^{(i)}-f(x^{(i)})\right)^2$ has a minimum value where $f:\mathbb R^d\to\mathbb R$ is a linear (an affine) function $f(x)=\beta_0+\beta_1x_1+\cdots+\beta_dx_d$ and where $\text{SS}_{\text{reg}}$ can be thought of as a function of $\beta_i$'s.)