Q:

calculus the following functions could you find the minimum of?

In finding the point on the line y=x closest to the point (0,1) which of the following functions could you find the minimum of?
a. $2x^2+2x+2$
b. $x^2-2x+2$
c. $2x^2-2x+1$
d. $2x^2-x+2$
e. $x^2-1$
Anyone can explain?

A:

If the $d$ is the distance, $$d^2=(0-x)^2+(1-y)^2=(0-y)^2+(1-y)^2=2y^2-2y+1=2x^2-2x+1$$
As $d>0,$  $d^2$ will be minimum  $\iff d$ is minimum

