Minimum d occurs at minimum d^2

[brycenrg]

Homework Statement

Find the point on the parabola y^2 = 2x that is closest to the point (1,4)

Homework Equations

d = ((x-1)^2 + (y-4)^2)^1/2
x=y^2/2

The Attempt at a Solution

My attempt involves subbing in x in the d equation and differentiating it. I can't get the same answer as the book because they differentiate d^2 = not d = to make it easier to solve.
They state "you should convince yourself the minimum of d occurs at the same point as the minimum of d^2, but d^2 is easier to work with"

How do i know it occurs at the same point?

 

[PeroK]

Try to find positive ##d_1 < d_2## with ##d_1^2 > d_2^2##

 

[LCKurtz]

Homework Statement

Find the point on the parabola y^2 = 2x that is closest to the point (1,4)

Homework Equations

d = ((x-1)^2 + (y-4)^2)^1/2
x=y^2/2

The Attempt at a Solution

My attempt involves subbing in x in the d equation and differentiating it. I can't get the same answer as the book because they differentiate d^2 = not d = to make it easier to solve.
They state "you should convince yourself the minimum of d occurs at the same point as the minimum of d^2, but d^2 is easier to work with"

How do i know it occurs at the same point?

If ##0<a<b## then ##a\cdot a < a \cdot b < b\cdot b##.