Find Shortest Distance of y=x^2 from (4,0)

[rhey]

shortest distance??

Homework Statement

find the shortest distance of y=x^2 from (4,0)

Homework Equations

The Attempt at a Solution

if y=x^2, then
y'=2x

where x=4
the slope is 8

the solution is

y=8(x-4)
if that is the tangent line \, the normal line would be

y=(1/8)(x-4)

after that.. I'm stuck!

 

[EnumaElish]

One way is to write out the Euclidian distance from vector [x, x^2] to vector [4, 0], then minimize w/r/t x.

 

[HallsofIvy]

EnumaElish, what rhey is trying to do is, I believe, more fundamental and a "nicer" method.

The shortest distance from (4,0) to y= x² will be along the line that is perpendicular to the graph. HOWEVER, rhey, x= 4 is for the point (4, 0), not any point on the graph! Let $(x_0, x_0^2)$ be the point on y= x² closest to (4, 0). The derivative there, and the slope of the tangent line, is 2x⁰ so the slope of the normal line is $-1/(2x_0)$. The equation of the line through (4, 0), perpendicular to y= x² at $(x_0,x_0^2)$ is $y= -1/(2x_0)(x- 4)$. If that is to pass through $(x_0, x_0^2)$, you must have y= $x_0^2= -1/(2x_0)(x_0- 4)$. Solve that equation for x₀ and you are almost done!

Hmm, that leads to a rather difficult equation for x₀! Darn it, EnumaElish may be right!

 

[rhey]

i really can't understand what are u trying to tell me.. honestly, I'm not that good in math! but I'm trying..

 

[colby2152]

1) Setup an equation for distance between point (0,4) and the function, $$f(x) = x^2$$

$$d = \sqrt{(x_2 - x_1)^2 - (y_2 - y_1)^2}$$
$$d = \sqrt{(x_2 - 4)^2 - (y_2)^2}$$
$$d = \sqrt{(x_2 - 4)^2 - (x_2)^4}$$

2) Take the derivative of d and set it equal to zero

$$d' = 0 = \frac{4(x_2)^3 - 2(x_2 - 4)}{2\sqrt{(x_2 - 4)^2 - (x_2)^4}}$$
$$0 = 4(x_2)^3 - 2(x_2 - 4)$$
$$4(x_2)^3 = 2(x_2 - 4)$$
$${x_2}^3 = x_2 - 4$$

As others have said, the problem is still difficult to solve from here...

 

[HallsofIvy]

And, as EnumaElish said, it's far easier to minimize the square of the distance, not the distance itself! Oh, and it should be y- 4, not x- 4.