Ok, here is an annoying one?

Question

I have been trying to figure this one out with a friend for about 3 days. We think we know the answer, but not sure how to get it mathematically

Find the point on the curve y=x^3 + 2x^2 - 5x - 6 that is closest to (0,0).

Could you please show your work if you understand this, and explain it please! Thanks

yeah, we have, and we think the closest point is (-1,0)

chulisoy, Thank you so much! I totally forgot about the distance formula

Yeah, I can not figure out the critical points. I have no idea how to solve a degree 6 polynomial

Answer 1

first find the distance from 0 to a point in the curve
(x, x^3 + 2x^2 - 5x - 6)
d((0,0), (x, x^3 + 2x^2 - 5x - 6) )
=sqrt( x^2+ (x^3 + 2x^2 - 5x - 6)^2)
since we want the minimum we can forget about the square root
and work with x^2+ (x^3 + 2x^2 - 5x - 6)^2
now the critical points of this function are:
2x+2( x^3 + 2x^2 - 5x - 6)(3x^2+4x-5) = 0
once you find the critical points, you can use the second derivative to decide whether they are local min or local max.
since it is a polinomial of degree 6 the one we are trying to find a minimum, then one of the local minima will work since the
lim x->+-infinity x^2+ (x^3 + 2x^2 - 5x - 6)^2 is infinity.

2x+2( x^3 + 2x^2 - 5x - 6)(3x^2+4x-5) =
6x^5 + 20x^4 - 24x^3 - 96x^2 + 4x + 60 = 0
we can divide by 2 and get:
3x^5 + 10x^4 - 12x^3 - 48x^2 + 2x + 30 = 0

which does not have integer roots... so it has be done by approximation.
Using maple i found the following aproximations:
51/64, 1019/512, -997/1024, -2233/1024, -95/32,

Answer 2

To solve this problem we need to find an equation that represents the distance from the origin.

d = sqrt(x^2 + y^2)
d = sqrt(x^2 + (x^3 + 2x^2 - 5x - 6)^2)
d = sqrt(x^2 + (x^3 + 2x^2 - 5x - 6)^2)
d = sqrt(x^6 + 4x^5 - 6x^4 - 32x^3 + 2x^2 + 60x + 36)

Now find the lowest possible value of d. The easiest way to find this value is to find the lowest possible value of the expression within the sqrt. x^6 + 4x^5 - 6x^4 - 32x^3 + 2x^2 + 60x + 36. This can be done by taking its derivative and solving for 0.

dy/dx = 6x^5 + 20x^4 - 24x^3 - 96x^2 + 4x + 60
6x^5 + 20x^4 - 24x^3 - 96x^2 + 4x + 60 = 0

Now the fun part comes in. Since there is no set way to factor a fifth degree polynomial, we will either need to guess a lot, or use a computer. I prefer the later.

My TI-92 Plus gives me these results x = 1.99102, 0.797401, -0.973308, -2.1802, -2.96824
These are the points where the slope of the equation is 0, or in other words, one of these points must be our minimum distance. Now we just plug these values into our original equation d = sqrt(x^2 + (x^3 + 2x^2 - 5x - 6)^2) and get the values of d to determine which one is the lowest.

x = 1.99102 -> d = 1.41739
x = 0.797401 -> d = 8.24692
x = -0.973308 -> d = 0.986509
x = -2.1802 -> d = 4.59466
x = 2.96824 -> d = 2.98444

The winner is x = -0.973308
Solve for y = -0.160845

(-0.973308, -0.160845)

Answer 3

when x=-1, y=0

so (-1,0) is a point on the curve.

Answer 4

Go do yur own home work k?