calculus
posted by mary on .
let f be the function
f(x) = x^3 + 3x^2  x + 2
a. the tangent to the graph of f at the point P = (2,8) intersects the graph of f again at the point Q. Find the coordinates of point Q.
b. Find the coordinates of point R, the inflection point of the graph of f
c. Show that the segment QR divides the region between the graph of f and its tangent at P into two regions whose areas are in the ratio of 16/11

let's first find the equation of the tangent at (2,8)
(you should have checked that (2,8) actually lies on f(x) )
f'(x) = 3x^2 + 6x  1
at the point given
f'(2) = 12  12  1 = 1
so y = x + b, but (2,8) lies on it
8 = 2 + b
b = 6
the tangent is y = x + 6
let's now intersect that with f(x)
x^3 + 3x^2  x + 2 =  x + 6
x^3 + 3x^2  4 = 0
we know that (x+2) is a factor since x = 2 was a solution
by synthetic division
x^3 + 3x^2  4 = (x+2)(x^2 + x  2)
x^3 + 3x^2  4 = (x+2)(x+2)(x1)
so we have a double root at x = 2 (this always happens at the tangent contact point)
and x = 1
when x = 1, y = 5
so Q must be (1,5)
b) I don't quite understand the question, but perhaps you can take it from here 
Find the coordinates of all points on yhe graph of y= 1x^2 at which the tangent line passes through the point (2,0).