Posted by **Mandy** on Friday, November 2, 2007 at 7:52am.

Find the point on the curve y=x^2 closest to point (0,1)

Here's what I have:

Sqrt {(x-0)^2 + ((x^2)-1)}

Sqrt {((x^2) + (x^2)-1}

d= sqrt (x^4)-1

y= (x^4)-1

Fprime(x) = 4x^3

Domain = ARN

Critical Number x= 0

Any help or comments would help alot. Thanks!

- Calculus I -
**Reiny**, Friday, November 2, 2007 at 8:24am
Mandy, you messed up in your first line

you forgot to square the last term in the brackets

dist = √((x-0)^2 + (y-1)^2)

=√(x^2 + (x^2 - 1)^2)

=√(x^4 - x^2 + 1)

then d(dist)/dx = 1/2(x^4 - x^2 + 1)^(-1/2)(4x^3 - 2x)

if we set this to zero we get

4x^3 - 2x = 0

for x=±1/√2 sub back to get y = 1/2

so the point is (1/√2,1/2) in the first quadrant

there is a second point in the second quadrant which is the same distance away because of the symmetry of the parabola.

- Calculus I -
**Reiny**, Friday, November 2, 2007 at 8:36am
there is an easier way to do these kind of questions.

It uses the fact that at the "closest" point the tangent must be perpendicular to the line from the given point to that tangent

So let the point on the curve be (a,a^2)

form y=x^2

dy/dx = 2x, so at (a,a^2) the slope of the tangent is 2a

the slope of the line from (0,1) to the tangent is then (a^2 - 1)/a

from the perpendicular line slope property

(a^2 - 1)/a = -1/(2a)

to get a = ±1/√2 as above

(notice that we also get a=0, which is correct according to the condition we imposed on the equation, namely that the lines had to be perpendicular.)

- Calculus I -
**drwls**, Friday, November 2, 2007 at 8:32am
You started out doing it correctly, but made an algebraic error.

d^2 = x^2 + (x^2-1)^2

d is a minimum where d^2 is a minimum, so you don't have to deal with the square root in finding the minimum distance.

d/dx(d^2) = 2x + 2(x^2 -1)* 2x

= 2x +4x^3 -4x = 4x^3 -3x

Set that = 0 and solve for x

One solution is x=0, where y=0

Another solution is

Another solution is where

4x^2 - 3 = 1

x = sqrt (3/4) = +or- 0.866

y = 0.75

Not all of these three points represent minima. Try them out (compute d^2) or use the second derivative test to find the true minimum. I get the minimum d^2 to be at

x = sqrt (3/4) = +or- 0.866

where y = 0.75

- Calculus I -
**drwls**, Friday, November 2, 2007 at 8:35am
I seem to disagree with some of the others above, so check themn all for accuracy.

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