examday

/TMUA

It is given that $f(x)=x^3+px+q$, where $p$ and $q$ are real constants, and that the equation $f(x)=0$ has three distinct real roots.

Which of the following statements is/are necessarily true?

• I $p>0$
• II The graph of $y=f(x)$ has a local maximum point with a positive $y$-coordinate.
• III The equation $f(x)=0$ has a rational root.