Consider the given the graph of $f^\prime(x)$ above. There is a local maximum on $f^\prime(x)$ when $x = -\cfrac {5}{3}$. Which of the following statements must be true?

I.$f$ is increasing for $x>-3$.

II.$f$ is concave down for $x>1$.

III.$f$ has a relative minimum at $x= 1$.

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