examday

/TMUA

$f(x)$ is a polynomial with real coefficients.

The equation $f(x)=0$ has exactly two real roots, $x=-p$ and $x=p$, where $p>0$.

Consider the following three statements:

• 1 $f'(x)=0$ for exactly one value of $x$ between $-p$ and $p$.
• 2 The area between the curve $y=f(x)$, the $x$-axis, and the lines $x=-p$ and $x=p$ is given by $2\int_0^p f(x)\,\dd x$.
• 3 The graph of $y=-f(-x)$ intersects the $x$-axis at the points where $x=-p$ and $x=p$ only.

Which of the above statements must be true?