Two points on a curve that have a common tangent line I have a curve $y = x^3 - x^4$ with an oblique common tangent line across the top of the curve. What is the equation of the tangent? 
Have spent hours on this but getting no-where.

Generally, $y=x^3-x^4=ax+b$ have 4 roots for $x$.
$$x^3-x^4 = ax+b \Rightarrow (x-x_1)(x-x_2)(x-x_3)(x-x_4) = 0 $$
If $ax+b$ is a tangent line on 2 points of the curve, that means 
$$x^3-x^4 = ax+b \Rightarrow (x-x_1)^2(x-x_2)^2 = 0 $$
Then we can have
$$ 
x^4 - x^3 +0\cdot  x^2 + ax + b = 0 
\Rightarrow 
$$
$$
x^4 - 2(x_1+x_2)x^3 + (x_1^2+x_2^2+4x_1x_2)x^2 - 2x_1x_2(x_1+x_2)x + x_1^2x_2^2 = 0
$$


*

*$2x_1 + 2x_2 = 1$

*$x_1^2 + 4x_1x_2 + x_2^2 = 0$

*$-2x_1x_2(x_1+x_2) = a$

*$x_1^2x_2^2 = b$
not too hard to find $a=\frac{1}{8}$, $b=\frac{1}{64}$

