Which Real Numbers Intersect This Curve at Four Distinct Points?

[anemone]

Find the real numbers $c$ for which there is a straight line that intersects the curve $y=x^4+9x^3+cx^2+9x+4$ at four distinct points?

 

[MarkFL]

My solution:

Spoiler

If we look at the concavity of the function, we see that we require the second derivative of the function to have two real, distinct roots:

\(\displaystyle f''(x)=12x^2+54x+2c\)

Requiring the discriminant to be positive gives us:

\(\displaystyle 54^2-4(12)(2c)>0\)

\(\displaystyle c<\frac{243}{8}\)

 

[anemone]

Hi MarkFL,

Thanks for participating and your answer is correct!(Clapping)