Derivative word problem. Tricky

[phil ess]

Homework Statement

A spaceship moves along y = x³ - 8x in the positive x-direction. Shutting off the engines at P allows it to move off along the tangent at P. Find P so that the ship can reach the point (4,0).

Homework Equations

Dunno

The Attempt at a Solution

So I start with P = (x₁,y₁). The slope of the tangent is then m = 3x₁² - 8 from the first derivative.

Then I need P, so I went like this:

m = (y₂ - y₁) / (x₂ - x₁) = 3x₁² - 8 where y₂ and x₂ are (4,0).

But now I have 1 Eqn. and 2 variables so I can't solve. I was thinking of using y = mx + b because I have y and x, plus an equation for m but then I end up with 2 equations and 3 variables because of the b term. Please help!

 

[HallsofIvy]

Okay, $m= 3x_1^2- 8$ and you know, I presume, that a straight line (except vertical) can be written y= m(x-a)+ b where m is the slope and (a, b) is a point on the line. What is the equation of the line with slope $3x_1^2- 8$ through (4, 0)?
You want that straight line to touch the graph when x= x1 so set the two equations, for the cubic and for the straight line equal and set x= x1. Solve for x1.

 

[phil ess]

Ok! I got it! Thanks a bunch