Let $f$:$RR$ be the function defined by $f(x)=a{x}^3+b{x}^2+cx+d.$

$($a$)$ Under what conditions on the coefficients $a,b,c,d$ will $f$ have one local minimum and one local maximum.

$($b$)$ Under what conditions on the coefficients $a,b,c,d$ will $f$ have a single local extremum.

$($c$)$ Under what conditions on the coefficients $a,b,c,d$ will $f$ have no local extrema $($i$.$e$.$ no local minima or maxima$).$

For this question you can use that a polynomial $p(x)=A{x}^2+Bx+C$ has two distinct roots if and only if $B^2-4AC>0$ and has exactly one root if and only if $B^2-4AC=0.$