Calculus Problem?

Question

Find the two points on the curve y = x^4 - 2x^2 - x that have a common tangent line.

Answer 1

Let A=(x₁ , y₁) and B=(x₂ , y₂) are the two points.
Slope of the tangent line is
(y₂-y₁)/(x₂-x₁)
Derivative of the function is
y' = 4x^3 - 4x - 1
Point A is on the curve:
y₁= x₁^4 – 2x₁^2 – x₁
Point B is on the curve:
y₂= x₂^4 – 2x₂^2 – x₂
Derivative in point A is equal to slope:
4x₁^3 - 4x₁– 1 = (y₂-y₁)/(x₂-x₁)
Derivative in point B is equal to slope:
4x₂^3 - 4x₂– 1 = (y₂-y₁)/(x₂-x₁)
The solution of the above four equation is:
x₁= -1
x₂= 1
y₁= 0
y₂= -2
The points are:
A = (-1, 0)
B = (1, -2)

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