Find the shortest distance from (3,0) to a point on the curve y=x^2-2x?

Question

How did you find the x-value of the derivative, 2.17, without estimating by looking at the graph?

Answer 1

We want to minimize distance d from a point (x,y) to (3,0) where x and y satisfies the equation y = x^2 - 2x. This will be equivalent to minimizing the distance squared (i.e. d^2).

So using the distance formula, we get that:
d^2 = (x - 3)^2 + (y - 0)^2
= (x - 3)^2 + (x^2 - 2x)^2
= x^2 - 6x + 9 + x^4 - 4x^3 + 4x^2
= x^4 - 4x^3 + 5x^2 - 6x + 9

Relabel this function to be f(x) = x^4 - 4x^3 + 5x^2 - 6x + 9

Goal: Finding the minimum value of this function.

First, we need to determine where minimum occurs. This can be done by finding the derivative and setting it equal to 0.

f'(x) = 4x^3 - 12x^2 + 10x - 6 = 0

Solving, we get that x = 2.17.

Notice that f''(x) = 10 - 24x + 12(x^2).
Plugging, f''(2.17) = 10 - 24(2.17) + 12(2.17^2) = 14.43 > 0
By the second derivative test, we see that x = 2.17 gives a minimum value for f, and hence d (after you take a square root).

So d^2 = f(2.17) = 0.82

And so, d = Sqrt(0.82) = 0.91

Shortest distance is approximately 0.91.

Answer 2

Slope = 2x-2 = 2a-2 at some point when x = a on the parabola.
So slope of line must be -1/(2a-2)
Equation of line is y = - x/(2a-2) +b
0 = -3/(2a-2) + b
So b = 3/(2a-2)
Slopes must be negative reciprocals so
3/(2a-2) = -1/(2a-2)
-2a +2 = 6a-6
8a=8
a=1
When x=a =1, y = -1
Hence we need to find the distance from 3,0 to (1,-1).
d = sqrt[(-1-3)^2 + (1-0)^2]= sqrt(17)

Answer 3

By sketching the graph of the equation, you know that the shortest distance is between the x values 2 and 3 of the equation.

Im still thinking of how you can use this, il edit my answer once i've worked it out

Answer 4

We can attack this problem by minimizing the distance from the point to the curve.

Distance^2 = D^2 = (x-3)^2 + (x^2-2x-0)^2

D^2 = x^2-6x+9+x^4-4x^3+4x^2

D^2 = x^4-4x^3+5x^2-6x+9

Now we can take the derivative of this and set it equal to 0 to find the x-value we desire

d/dx = 4x^3-12x^2+10x-6=0

Solving for x (there is only one real root) gives
x=2.165373043

Plugging into y=x^2-2x gives y=.3580943294

Now we can compute the distance between the two points as Sqrt((3-2.165373043)^2 +(0-.3580943294)^2).

Simplifying gives the answer, 1.22.

Answer 5

Why?