Question

Find the point on the curve y=x^(1/2) that is a minimum distance from the point (4,0).

My book says you use the distance formula.

Then you let L = D^2 because the minimum value of D^2 will occur at the same value of x as the minimum value of D.

What is L, though.

Answer 1

L is the distance^2. You don't have to do that way, as I will demonstrate.

D^2=(4- x)^2+(0-y)^2 that comes from the distance formula.

Doing it the way the L=D^2 did:

L= ..
dL/dx=0=2(4-x)+2(y)dy/dx

but dy/dx = d(sqrt x)/dx= 1/2sqrtx
so 0=-2x+2sqrtx/2sqrtx or
2x=2
x= 1/2, y= 1/sqrt2

Now, lets do it without the L substitution:
D^2=(4- x)^2+(0-y)^2 that comes from the distance formula.

2D dD/dx=0=2(4-x)+2(y)dy/dx
again, dy/dx= d(sqrtx)/dx= 1/(2sqrtx)
so 0=-2x+2sqrtx/2sqrtx
and again x=1/2, y= 1/sqrt2

Answer 2

They are saying, let D^2 = L
so when later on you differentiate
the result for L is simpler than that for D^2

They are using the property that if a > b
then a^2 > b^2.

let the closest point be P(x,y)
then
L = D^2 = (x-4)^2 + (y-0)^2
= (x-4)^2 + (x^(1/2))^2
= (x-4)^2 + x
dL/dx = 2(x-4) + 1 = 0 for a min distance
2x - 8 + 1 = 0
x = 7/2

if x=7/2 , then y = √(7/2) = √7/√2 = √14/2

the closest point is ((7/2 , √14/2)