How can someone find the points of tangency of a parabola in this situation?
I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis is an equilateral triangle.
What approach should I take?
I know that I can find the slope of a tangent line passing through the point (x,y) by just taking the derivative of that function and by using the point-slope formula I can get the equation of that tangent line. However, I would need to know the point...
Hint: In order to form an equilateral triangle with the $x$ axis, one of the tangents must slope upwards by 60° and the other must slope downwards by 60°. But as long as they do this (and don't happen to have the same $x$ intercept) they will form an equilateral triangle somewhere.