Let

1. Prove that $x=a$ is both a root and a stationary point of the cubic curve $y=f(x).$

2. Prove that $x=a$ gives a minimum of $f(x)$ if $a>b,$ and a maximum point if $a<b.$ Sketch the curve in both cases.

(This finally explains why a repeated root of a cubic just touches the $x$ axis, something that we had only assumed until now.)

Expand and differentiate. Demonstrate that $x=a$ is a root of both $f(x)$ and $f^{\prime }(x).$

For (b), you should find that

In order to sketch the cubics, notice that the coefficient of $x^3$ in $f(x)$ is $1,$ which is positive, so this is an up-down-up cubic.

You can slide $a$ and $b$ around in this picture to help: