Shipping Derivatives

A spaceship moves along y = x ^ 3 - 8x in positive x direction. Turn off the engine at P allows it to move along the tangent at P. Find P so that the ship can reach the point (4, 0).

I understand that y '= 3x ^ 2-8

SK response that will not work because the point (4,0) is not on the original curve. If you draw a picture of the problem with the spacecraft firing off tangent to the curve until it touches (4.0), you'll see that.

So what you have to find a line that passes through (4,0), has a slope of 3 x ^ 2 to 8 at this location x, and also passes the original curve at this point x.

You know a line has an equation like y = mx + b, where m is the slope. You know that (4.0) is one of the points on this line. You know the other point, (p, b), must satisfy the equation y = x ^ 3 - 8x. And you know that the slope of this line (y / delta x delta) must be equal to 3x ^ 2 to 8. You have 2 equations (b = 3 ^ P - 8P and (0-b) / (4-P) = 3b ^ 2 to 8). Then you can solve for 2 unknowns, P and B.

use rules dy dx / and set equal to p
and then the answer is here!

My bad.