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From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf:

question 8(a), (i)-(iv)

Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x > 0$.
(i) Show that the minimum value of $f(x)$ occurs when $x=\frac{a+b}2$.
(ii) Suppose that $c$ is a positive real number.
Show that $\left(\frac{a+b+c}{3\sqrt[3]{abc}}\right)^3 \ge \left(\frac{a+b}{2\sqrt{ab}}\right)^2$ and deduce that $\frac{a+b+c}3\ge\sqrt[3]{abc}$.
You may assume that $\frac{a+b}2\ge\sqrt{ab}$.
(iii) Suppose that the cubic equation $x^3-px^2+qx-r=0$ has three positive real roots. Use part (ii) to prove that $p^3\ge27r$.
(iv) Deduce that the cubic equation $x^3-2x^2+x-1=0$ has exactly one real root.

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At the very least, copy the problem here for us to see... –  Mariano Suárez-Alvarez Feb 5 '11 at 6:31
Also, it is always a good idea to tell what you have done so far, what the problem is, and in what context you're doing this. This looks like homework. –  Fredrik Meyer Feb 5 '11 at 6:43

1 Answer 1

(i) Minima of a function occur at critical points—where the derivative of the function is undefined or zero. Have you found the critical points of the function?

(ii) Are you stuck on the "Show that..." part, or the "... deduce that..." part?

(iii) If the three positive real roots are $a$, $b$, and $c$, then $x^3-px^2+qx-r=(x-a)(x-b)(x-c)$—what are $p$ and $r$ in terms of $a$, $b$, and $c$? How do these relate to (ii)?

(iv) For a polynomial with real coefficients, any nonreal complex roots occur in conjugate pairs. Consequently, for a cubic polynomial, there must be exactly __ or __ real roots (fill in the blanks).

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