The function $f$ is defined by

1. Prove that $f$ has a local minimum when $x=2.$

2. Hence, explain why $f^{-1}$ does not exist.

For (a), you must prove that $x=2$ gives a turning point using $f^{\prime }$, and prove that this is a minimum point using $f^{″}$.

For (b), what does the curve look like near the local minimum? Can you explain why it must be many-to-one? It would be helpful if your explanation included a sketch.