What Is the Smallest k for a Cubic Function to Have Exactly One Real Root?

[anemone]

Here is this week's POTW:

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Compute the smallest value $k$ such that for all $n>k$, the cubic function $x^3+x^2+nx+9$ has exactly one real root.-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

No one answered last week's problem.(Sadface)

You can find the suggested solution below:

Spoiler

In order to compute the smallest value $k$ such that for all $n>k$, the cubic function $x^3+x^2+nx+9$ has exactly one real root, we first let $f(x)=x^3+x^2+nx+9$ has a negative root $a$ and a double root $b$.

By the Vieta's formula, we have:

$ab^2=-9$

$a+2b=-1$

Solving them for $b$, we get

$2b^3+b^2-9=0\\ \therefore (2b-3)((b+1)^2+2)=0$

This means $b=\dfrac{3}{2}$ is the only real solution of $f(x)$ so $a=-4$.

This gives $p=-\dfrac{39}{4}$.