Real roots and derivative polynomial

Let $P$ be a monic polynomial whose roots are all real. Are the roots of its derivative $P'$ also real ?

Conversely, if the roots of $P'$ are all real, does it imply that the roots of $P$ are real too ?

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Solution by Anduril

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For the first part, use Rolle’s Theorem and the fact that if a is a root of P with the order p then it is a root of P' of order p-1 so if we have the roots $a_{1}$ ,..., $a_{m}$ with multiplicities
$n_{1}$ ,..., $n_{m}$ , we have m-1 roots for P' that are not those of P with Rolle and $n_{i}$ -1 from each root $a_{i}$ of P so in total
$\sum_{i=1}^{m} (n_i - 1) + m - 1 = \left( \sum_{i=1}^{m} n_i \right) - 1 = \deg(P) - 1 = \deg(P')$

The second point is wrong, just consider $X^{2}$ +1 for P.

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