SSCE HSC Mathematics Extension 2 Proofs involving inequalities: Suppose that $a$ and $b$ are pos

Question

Suppose that $a$ and $b$ are positive real numbers, and let $f(x) = \frac{a + b + x}{3(abx)^{\frac{1}{3}}}$ 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 $ \frac{(a + b + \frac{c}{3})}{\sqrt{abc}} \geq \frac{2(a + b)}{\sqrt{ab}} $ and deduce that $ \frac{a + b + c}{3} \geq \frac{2}{3 \sqrt{abc}}. $

You may assume that $\frac{a + b}{2} \geq \sqrt{ab}.$

(iii) Suppose that the cubic equation $x^3 - px^2 + qx - r = 0$ has three positive real roots. Use part (ii) to prove that $p \geq 2\sqrt{q}.$

(iv) Deduce that the cubic equation $x^3 - 2x^2 + x - 1 = 0$ has exactly one real root.

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The point $P(\sec \theta, b \tan \theta)$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$ The line through $P$ perpendicular to the $x$-axis meets the asymptotes of the hyperbola, $ \frac{y}{a} - \frac{x}{b} = 0 $ and $\frac{y}{b} - \frac{x}{a} = 0,$ at $A(\sec \theta, b \sec \theta)$ and $B(\sec \theta, -b \sec \theta)$ respectively. A second line through $P$, with gradient $\tan \alpha$, meets the hyperbola at $Q$ and meets the asymptotes at $C$ and $D$ as shown. The asymptote $\frac{y}{a} - \frac{x}{b} = 0$ makes an angle $\beta$ with the $x$-axis at the origin, as shown.

(i) Show that $AP \times PB = b^2.$

(ii) Show that $CP = AP \cos \beta$ and that $PD = \frac{PB \cos \beta}{\sin(\alpha - \beta)}.$

(iii) Hence deduce that $CP \times PD$ depends only on the value of $\alpha$ and not on the position of $P.$

(iv) Let $CP = p, CP = r.$

Show that $p = q.$

(v) A tangent to the hyperbola is drawn at $T$ parallel to $CD$. This tangent meets the asymptotes at $U$ and $V$ as shown. Show that $T$ is the midpoint of $UV$.

Suppose that $a$ and $b$ are positive real numbers, and let $f(x) = \frac{a + b + x}{3(abx)^{\frac{1}{3}}}$ for $x > 0.$ (i) Show that the minimum value of $f(x)$ occurs when $x = \frac{a + b}{2}$ - HSC - SSCE Mathematics Extension 2 - Question 8 - 2005 - Paper 1

Worked Solution & Example Answer:Suppose that $a$ and $b$ are positive real numbers, and let $f(x) = \frac{a + b + x}{3(abx)^{\frac{1}{3}}}$ for $x > 0.$ (i) Show that the minimum value of $f(x)$ occurs when $x = \frac{a + b}{2}$ - HSC - SSCE Mathematics Extension 2 - Question 8 - 2005 - Paper 1

Show that the minimum value of f(x) occurs when x = a + b/2

Show that CP = AP cos(β) and that PD = PB cos(β)/sin(α - β)

Hence deduce that CP × PD depends on α and not on the position of P.

Show that p = q.

Show that T is the midpoint of UV.

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