Answer
(a) 3 x-intercepts and 2 local extrema
(b) 1 x-intercepts and 0 local extrema
(c) see explanations below.
Work Step by Step
(a) 3 x-intercepts and 2 local extrema
(b) 1 x-intercepts and 0 local extrema
(c) For $P(x)=x^3-ax$, there are 3 x-intercepts and 2 local
extrema, while for $P(x)=x^3a+x$, there is 1 x-intercepts and 0 local extrema.
Explanations:
$P(x)=x^3-ax=x(x^2-a)$ as $a>0$ there are three real solutions and the
curve pass the x-axis 3 times indicating two extrema.
However, $P(x)=x^3+ax=x(x^2+a)$ has only one solution at $x=0$ which means
the curve pass the x-axis only once and there will be no extrema.
