Question

Find the points on the curve $y=x^3-2x^2-x$ at which the tangent lines are parallel to the line $y=3x-2$.

Answer 1

$y=x^3-2x^2-x$.

$y^1={3x}^2-4x-1$

when this is parallel to $y=3x-2$, then slopes are equal

$\Rightarrow {3x}^2-4x-1=3$

$\Rightarrow {3x}^2-6x+2x-4=0\Rightarrow (3x+2)(x-2)=0$

$x=\frac{-2}{3},2$

$y=\frac{-14}{27},-2$