Further Problem on Tangents Requiring the Use of Differentiation

Question

Show that the tangent to the curve $y = x^3 - 2x$, at the point with $x$-coordinate $a$ meets the curve again at a point with $x$-coordinate $-2a$.

Answer 1

we have $$y(x)=x^3-2x$$ then $$y'(x)=3x^2-2$$ then $$y'(a)=3a^2-2$$ and the tangent line has the form $$y(x)=(3a^2-2)x+n$$ for $x=a$ we get $$y(a)=a^3-2a$$ and you can compute $n$: $$a^3-2a=(3a^2-2)a+n$$ can you finish?

Answer 2

Two steps:

• set up the equation of the tangent line at $(a,f(a))$;
• find the points of intersection between this line and $y=x^3-2x$.

Hint: remember that you already know that $x=a$ will be a solution; this may help in finding the other one.