Question

Determine where, if anywhere, the tangent line to f(x) = x^3 - 5x^2 + x is parallel to the line y = 4x +23.

Answer 1

can you find the derivative of f(x)?
if so, set the derivative equal to 4 (the slope of the line)
and solve for x

Answer 2

if we know that the first derivative of f(x) is the slope of the tangent, then all we need to do is set this Fprime equl to 4

Answer 3

@phi Why set the derivative equal to 4?

Answer 4

you want a line
parallel to the line y = 4x +23.

parallel means equal slopes. 4 is the slope of the given line (y = mx+b format)

as you may know, the derivative of a curve will give you the slope of the tangent line to the curve at any point.

Answer 5

@phi Then would I just simply simplify the equation and that will be the line?

Answer 6

after the derivative, you get a quadratic. solve for x. if you get real numbers, then that is the answer. if you get complex numbers, there is no tangent that is parallel to the given line.

Answer 7

I'm confused on the quadratic part of the equation.

Answer 8

did you get the derivative?

Answer 9

Yes, I got 3x^2 - 10x + 1

Answer 10

so set = to 4:
3x^2 - 10x + 1=4 or
3x^2 - 10x -3= 0

use the quadratic formula to solve.
http://www.purplemath.com/modules/quadform.htm

Answer 11

I have \[(10 \pm \sqrt{136} ) \div 6\]

Answer 12

so you have 2 x values where the tangent line will be parallel to the given line.
You can find the corresponding y value (use the original equation) to give 2
(x,y) pairs as the answer to the question.

Answer 13

@phi I understand now! Thank you very much!

Answer 14

@phi But would I use the f(x) equation or the y equation?

Answer 15

the f(x) equation (you are looking for a tangent to f(x), so the (x,y) point has to be on that curve)
the other line y= 4x+23 is parallel to the tangent , but not the same as the tangent line.

Answer 16

@phi Thank you!