What Is the Value of b If All Roots of the Polynomial Are Positive Real Numbers?

[anemone]

Here is this week's POTW:

-----

If all the roots of $x^4-8x^3+24x^2+bx+c=0$ are positive reals, find the value of $b$.

-----

 

[anemone]

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

However, I have decided to give the community another week to attempt at the problem.(Smile)

 

[anemone]

Congratulations to Opalg for his correct solution (Cool) , which you can find below:

Spoiler

If $f(x) = x^4-8x^3+24x^2+bx+c$ has four real roots then its derivative must have three real roots. But $$f'(x) = 4x^3 - 24 x^2 + 48x + b = 4(x-2)^3 + b + 32,$$ and the function $(x-2)^3$ is strictly increasing except at the point $x=2$. So $f'(x)$ can only have three real roots if $b+32 = 0$. Therefore $b = -32$.