# Find the Real Numbers c

#### anemone

##### MHB POTW Director
Staff member
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

Staff member
My solution:

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

##### MHB POTW Director
Staff member
Hi MarkFL,

Thanks for participating and your answer is correct!