Review question

# What can we say if $y = m(x-a)$ is tangent to $y = x^3 - x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8287

## Suggestion

The graphs of $y = x^3 - x$ and $y = m(x-a)$ are drawn on the axes below. Here $m>0$ and $a \le -1$.

The line $y=m(x-a)$ meets the $x$-axis at $A = (a,0)$, touches the cubic $y=x^3 -x$ at $B$ and intersects again with the cubic at $C$. The $x$-coordinates of $B$ and $C$ are respectively $b$ and $c$.

1. Use the fact that the line and cubic touch when $x=b$, to show that $m=3b^2-1$.

What do we know about curves and lines which touch at some point? How can that help us to answer this question?

1. Show further that $a = \frac{2b^3}{3b^2 - 1}.$

Can we find $m$ simply in terms of $a$ and $b$? Could we drop a vertical from $B$?

1. If $a=-10^6$, what is the approximate value of $b$?

Can you sketch a graph to show this situation?

1. Using the fact that $x^3 - x - m(x-a) = (x-b)^2 (x-c)$ (which you need not prove), show that $c=-2b$.

Why is this true? For which values of x is this true? So this tells us that …

1. $R$ is the finite region bounded above by the line $y=m(x-a)$ and bounded below by the cubic $y = x^3 - x$. For what value of $a$ is the area of $R$ largest?

Show that the largest possible area of $R$ is $\dfrac{27}{4}$.

How does the area of $R$ change as $a$ changes?

Or… can we find a formula for the area? When we’ve done so, how can we maximise this?