Question

For what values of x is the tangent line of the graph of
f(x)=2x^3−12x^2+19x+0
parallel to the line y=1x+0.7 ? Enter the x values in order, smallest first, to 4 places of accuracy:

Answer 1

the slope of the line is 1. So, you want the slope of the tangent to be 1.

f'(x) = 6x^2-24x+19 = 1
x = 1,3


check:
f(1) = 9
f(3) = 3

So the tangent lines are
y-9 = x-1, or y=x+8
y-3 = x-3, or y=x

the graphs are at

http://www.wolframalpha.com/input/?i=plot+y%3D2x%5E3%E2%88%9212x%5E2%2B19x,+y%3Dx%2B8,+y%3Dx