342,864 views
29 votes
29 votes
The curve with equation y = x³ +3 has two tangents parallel to the line with equation

y = 12x-1. Find the co-ordinates of the two points.

The curve with equation y = x³ +3 has two tangents parallel to the line with equation-example-1
User Shaquille
by
2.4k points

1 Answer

16 votes
16 votes

Answer:

(-2, -5), (2, 11)

Explanation:

You want the coordinates of the points on the curve y = x³ +3 where the tangent lines are parallel to y = 12x -1.

Slope

The slope of the tangent line is the x-coefficient in its equation: 12.

The slope of the curve is given by its derivative:

y' = 3x²

We want the x-values where the slope is 12. These are the solutions to ...

12 = 3x²

4 = x² . . . . . . . . . . divide by 3

x = ±√4 = ±2 . . . . . take the square root

Coordinates

The coordinates of the points with x = ±2 are ...

y = (±2)³ +3 = ±8 +3 = {-5, 11}

The tangent points are (-2, -5) and (5, 11).

__

Additional comment

The attached graphs and table show the solutions to y'=0 and the corresponding point locations. It also shows the tangent line equations (in point-slope form).

The curve with equation y = x³ +3 has two tangents parallel to the line with equation-example-1
User Alex Mantaut
by
3.0k points