Maximum number of possible intersections between tangent line and function $x^3$ I'm struggling with a problem but can't find a way how to solve it:
Calculate maximum number of possible intersections between tangent line of function $f(x) = x^3$ and function $f(x) = x^3$ where $x \in \mathbb{R}$.
I know I should use derivatives but I don't have a clue how.
 A: Hint:
An equation for the tangent line at the point $(a, f(a))$ for any differentiable function $f$ is given by the formula
$$f(x)=f(a)+f'(a)(x-a).$$
Other than that, you obtain here  a cubic equation in $x$, and the abscissa $a$ of the point of contact is a root of multiplicity at least $2$.
A: Pick a point $x_0$, and let $g(x)=3x_0^2(x-x_0)$ be the expression for the tangent of $f$ at $x_0$ (the actual expression for $g$ isn't actually that important).
We want the number of roots of the cubic equation $f(x)-g(x)=0$. We know $x=x_0$ is a root, because a tangent intersects the graph it is tangent to. However, we actually know that this is at least a double root, because the tangent is a tangent (said with derivatives, $x_0$ is a root of $(f(x)-g(x))'=0$ as well)
Since we have a cubic equation, there can be at most three solutions. And since at least two of them are at $x_0$, there is at most room for one more.
A: Hint:
After finding the derivative of $f(x)$, you can find the general form of the equation of the tangent line in terms of $x$. For now, let it be $y=mx+c$ Then the intersection of $f(x)$ and the tangent line would be $x^3=mx+c$. Then, you could use the theorem:
For cubic equation $x^3+px+q=0$, there are 3 solutions if and only if $4p^3+27q^2<0$.
