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

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    2017
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The discussion focuses on determining the smallest value of k for which the cubic function x^3 + x^2 + nx + 9 has exactly one real root for all n greater than k. Participants are encouraged to engage with the Problem of the Week (POTW) and contribute solutions. The thread highlights the importance of understanding cubic functions and their roots in relation to the parameter n. A suggested solution is provided, but no responses have been recorded for the previous week's problem. Engaging with these types of mathematical challenges can enhance problem-solving skills.
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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!
 
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No one answered last week's problem.(Sadface)

You can find the suggested solution below:

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 $p=-\dfrac{39}{4}$.
 
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