At what point on the graph of $y=-3x^3+2x-1$ is the tangent parallel to $y=2x+10$?
Now do I solve this question algebraically or do I solve it graphically since there is no specific x value given to find the slope of the tangent using the IROC method.
Best Answer
Since the tangent line is parallel to $y=2x+10$ then it's of the form $y=2x + d$ for some real number $d$. Now plug this into the equation and see when it obtains a double/triple real root.
$$2x + d = -3x^3 + 2x - 1 \implies -3x^3 = d+1 \implies x^3=\frac{d+1}{-3}$$
This equation has a real triple root only when RHS is $0$, as otherwise we get two complex roots (multiples of the third roots of unity), so therefore we get that $d=-1$ and they touch each other at $(0,-1)$