Investigating the real roots of a cubic equation

[brotherbobby]

Homework Statement

Show that has only one real root unless $-5\le k\le\dfrac{7}{4}$.

Relevant Equations

(1) An th order polynomial will have at least one real root if .
(2) Rolle's theorem : If is a polynomial equation and , then there exists at least one value in the interval such that .
(3) From Rolle's theorem (2) above, it follows that if our polynomial equation has three real roots, then between any consecutive pair of them there must be a real value of for which .
(4) The condition (3) above is necessary, but not sufficient. Sufficience is however guaranteed if it can be shown that for those two values of , say some , the values of the original function must have opposite signs. Thus .

Statement of the problem : Let me copy and paste the problem as it appears in the text. [Riley, K.F., Hobson,M.P., Bence,S.J. (2006) Mathematical Methods for Physics and Engineering. (3E)].

Attempt : One real root is guaranteed, as $n =\;\text{odd}$. Differentiating $f'(x) = 12x^2+6x-6=6(2x-1)(x+1)$. Thus $f'(-1) = f'(1/2) = 0$. Hence, the derivative $f'(x)$ itself has two real roots, which indicates that $f(x)$ may have three real roots.
To test whether it does, we use condition (4) in the Relevant Equations section, given above. Namely, evaluate the value of the function $f(x)$ at those two values where its derivative vanishes to see whether they are of opposite signs, i.e. whether $f(\beta_1)f(\beta_2)<0$, where $\beta_1 = -1,\;\beta_2=1/2$.
$\small{f(-1) = 5-k\;\text{and}\; f(1/2) = -7/4-k\Rightarrow (k-5)(k+7/4)<0\Rightarrow\boxed{\color{red}{-7/4\le k\le 5}}}$.

Text answer : This is different from the answer in the text, where $\boxed{\color{blue}{-5 \le k \le 7/4}}$.
The text answer is correct, as I could verify at $\verb|desmos.com|$.

Request : A hint as to where am I going wrong in my working.

Many thanks.

 

[pasmith]

Your working appears to be correct, and I obtained the same result by considering the discriminant of $f$.

What did you do to verify the text answer? According to the text, for $k = -3$, $f$ should have more than one real root, but according to Wolfram Alpha there is only one real root and a complex conjugate pair.

I must therefore conclude that the text answer is incorrect. It happens.

EDIT: It is a bit strange to have a minus sign before the $k$ in the problem statement, so maybe that was meant to be a plus sign, which would make the answer in the text correct.

 

[brotherbobby]

Your working appears to be correct, and I obtained the same result by considering the discriminant of $f$.

What did you do to verify the text answer? According to the text, for $k = -3$, $f$ should have more than one real root, but according to Wolfram Alpha there is only one real root and a complex conjugate pair.

I must therefore conclude that the text answer is incorrect. It happens.

EDIT: It is a bit strange to have a minus sign before the $k$ in the problem statement, so maybe that was meant to be a plus sign, which would make the answer in the text correct.

Hi @pasmith, yes it seems I was mistaken too in my plotting on $\verb|desmos.com|$. I took the constant $k$ with a positive $(+)$ sign.

Of importance is the minus ''$-$'' sign before the term $k$ in the question, as you pointed out. If it was $+k$, then the book is correct. As it stands, it's mistaken.

One small digression. Have you observed that here on PF, when you try to edit your thread (as the OP), the formatting in the Title, the Problem Statement and Relevant Equations all disappear and have to be typed in all over again. Can you ask someone to look into it?
I do not know any admins here in PF.

 

[Ibix]

One small digression. Have you observed that here on PF, when you try to edit your thread (as the OP), the formatting in the Title, the Problem Statement and Relevant Equations all disappear and have to be typed in all over again. Can you ask someone to look into it?
I do not know any admins here in PF.

@Greg Bernhardt

 

[brotherbobby]

Thank you @Ibix. I should add, to help the admin, @Greg Bernhardt, that the formatting in $\rm{\LaTeX}$ does not disappear in the main text, where we type in our attempt and doubts, if we try to edit our message. Needless to say, this is also not a problem in all future replies.
But for the first time that you post a thread as an OP, it's difficult to edit your posted message in case you made mistakes. All $\rm{\LaTeX}$ formulae and equations disappear from the Title, The Problem Statement (Homework) and the Relevant Equations.
Many thanks.

 

[Mark44]

All $\rm{\LaTeX}$ formulae and equations disappear from the Title, The Problem Statement (Homework) and the Relevant Equations.

When I looked at post #1 of your thread I saw that there was some malformed LaTeX in the Homework Statement section, namely the inequality at the end of the statement. I fixed this by inserting a pair of # characters at the beginning and end of the inequality.
I don't know whether you had written this part appropriately at first and the forum software deleted it, or whatever else you might have done.

 

[Greg Bernhardt]

Thank you @Ibix. I should add, to help the admin, @Greg Bernhardt, that the formatting in $\rm{\LaTeX}$ does not disappear in the main text, where we type in our attempt and doubts, if we try to edit our message. Needless to say, this is also not a problem in all future replies.
But for the first time that you post a thread as an OP, it's difficult to edit your posted message in case you made mistakes. All $\rm{\LaTeX}$ formulae and equations disappear from the Title, The Problem Statement (Homework) and the Relevant Equations.
Many thanks.

The latex doesn't render or the latex is cleaned?

 

[Mark44]

The latex doesn't render or the latex is cleaned?

The LaTeX in the Homework Statement section (and only there) was malformed owing to the absence of the pre- and post-tags. Whether it was stripped off by the forum software or wasn't there in the first place, I can't say. I added the missing tags to it and it displays now as it should.

 

[brotherbobby]

The latex doesn't render or the latex is cleaned?

The $\rm{\LaTeX}$ renders but is cleaned upon an edit. You have to type all LaTeX codes again.

 

[Ibix]

The $\rm{\LaTeX}$ renders but is cleaned upon an edit. You have to type all LaTeX codes again.

Note, for example, point 2 in "Relevant equations", where there clearly should be some maths. Presumably the LaTeX got swallowed at some point.

 

[brotherbobby]

Note, for example, point 2 in "Relevant equations", where there clearly should be some maths. Presumably the LaTeX got swallowed at some point.

None of the codes in the Relevant Equations show. It is my mistake too that when I decided to edit my original message that starts the thread (#1), I edited only the "body" of the message, without looking at the title or the Relevant Equations, or the Problem Statement. Anything written in $\rm{\LaTeX}$ disappears.
The best we can do in these situations is to write our threads perfectly, so that no editing is required.
The problem has been here for more than six months. It will be good if it was solved.
I can also confirm that the problem didn't exist until about six or eight months ago. @Greg Bernhardt

 

[Greg Bernhardt]

I can also confirm that the problem didn't exist until about six or eight months ago. @Greg Bernhardt

I'll take a look at this in depth today or tomorrow, thanks!

 

[Mark44]

Note, for example, point 2 in "Relevant equations", where there clearly should be some maths. Presumably the LaTeX got swallowed at some point.

Or not. My sense is that, more often than not, people do not include actual equations in this section.

 

[Greg Bernhardt]

I can confirm that on initial save it's fine, but on edit and new save, the section is cleaned of latex code. I'll look into this more.