Let f(x)=ax^3+bx^2+cx+d , find conditions for a,b.c and d so that
a) f has no horizontal tangent line.
b) f has aunique horizontal tangent line.
c) f has tow horizontal tangent lines
The discrimiant is
$\displaystyle (2b)^2-4(3a)(c)=4b^2-12ac=4(b^2-3ac)$
This will have no solutions when the discrimiant is negative.
This will have exactly one solution when the discrimiant is equal to zero.
This will have two solutions when the discrimiant is positive.