Posted by **kyle** on Sunday, January 16, 2011 at 10:47pm.

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.

- calculus -
**bobpursley**, Sunday, January 16, 2011 at 11:06pm
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

- calculus -
**Reiny**, Sunday, January 16, 2011 at 11:13pm
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)

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