SOLUTION: Find k and the points T of contact if the parabola y=x^2

SOLUTION: Find k and the points T of contact if the parabola y=x^2 - k touches the cubic y=x^3

Question 1145198: Find k and the points T of contact if the parabola y=x^2 - k touches the cubic y=x^3
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Find k and the points T of contact if the parabola y=x^2 - k touches the cubic y=x^3
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If by "touch" you mean tangent:
If k = 0, they are tangent at the Origin, and there is an intersection.
=============
y = x^3
y' = 3x^2 (1st derivative = slope at x)
----
y = x^2 - k
y' = 2x
==========
Find where the 2 slopes are equal.
3x^2 = 2x
3x^2 - 2x = 0
x = 0
=====
3x = 2
x = 2/3
---
(2/3)^3 - (2/3)^2 + k = 0
8/27 - 12/27 = -k
k = 4/27
========
And there's an intersection in Q3.

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