# Polynomial problem

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

(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)?