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Andrew J.

asked • 11d

Show that x^3+px+q=0 has: (a) one real root if p>0, and (b), three real roots if 4p^3+27Q^2<0

Hello,


Trying to answer the question:


Show that x3+px+q=0 has: (a) one real root if p>0, and (b) three real roots if 4p3+27q2<0


The 'tools' I have at my disposal for the question are the Intermediate Value Theorem, the Mean Value Theorem, Rolle's Theorem, local extrema, and increasing vs. decreasing functions (based on positive or negative derivative).



William W.

tutor
Your question is a little confusing because the first equation infers that "x" is the independent variable, however, your second equation has no "x" in it. Is there something written incorrectly?
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11d

Andrew J.

Thank you William. No, the question is accurate as written. I believe I've figured it out after watching some videos in Hindi on YouTube: For part a, my 'proof' is that, as per the equation given, there must be at least one real root for x^3+px+q to equal zero. Because the derivative, 3x^2+p is positive for all x when p>0, then there can only be one root because the function is increasing for all real numbers. (Does this sound correct?) For part b, I note that the function has three real roots if the function has exactly one local extremum where for which the function is less than zero and exactly one local extremum for which the function is greater than zero. Setting the derivative to zero gives f'(x)=+-sqrt(-p/3)=0. As per the above, the function thus has three real roots if f(sqrt(-p/3)) times f(-sqrt(-p/3))<0. If I solve the functions for each extremum and then multiply them together, I get 4p^3/27+q^2=4p^3+27q^2 which, as per the problem, is less than zero and thus, the function has 3 real roots. Please let me know if you see any flaws here! Warmly, Andrew
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11d

2 Answers By Expert Tutors

By:

Andrew J.

Thank you Richard! I believe this is the solution I finally arrived at and posted as a comment below my question above. Much appreciated!
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10d

Andrew J.

Thank you Paul. I haven't covered Cardan's solution yet, but it sounds like I'm still good to go given the requirement in the question for 3 real roots, which requires that -p be greater than zero.
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10d

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