1. ## Horizontal line

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

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
If you take the derivative you get

$\displaystyle f'(x)=3ax^2+2bx+c$

So a graph has a horizontal tangent when the derivative is equal to zero.

so we need to analyze the discriminant of the quadratic.

Can you finish from here?

3. Originally Posted by TheEmptySet
If you take the derivative you get

$\displaystyle f'(x)=3ax^2+2bx+c$

So a graph has a horizontal tangent when the derivative is equal to zero.

so we need to analyze the discriminant of the quadratic.

Can you finish from here?
How could I find the three conditions? thank you for help

$\displaystyle (2b)^2-4(3a)(c)=4b^2-12ac=4(b^2-3ac)$