examday

/TMUA

Let $f$ be twice differentiable on $\mathbb R$. Define

$P:$ $f$ has at least four distinct real roots,

$Q:$ $f'$ has at least three distinct real roots,

$R:$ $f''$ has at least two distinct real roots.

Which statements are true?

• I. $P$ only if $Q$.
• II. $Q$ only if $R$.
• III. $R$ only if $P$.