Question
If the arithmetic mean of two positive numbers $a$ and $b(a>b)$ is twice their geometric mean, then $a:$ $b$ is(a) $2+\sqrt{3} ; 2-\sqrt{3}$(b) $7+4 \sqrt{3}: 1$(c) $1: 7-4 \sqrt{3}$(d) $2: \sqrt{3}$
Step 1
This can be written as: \[\frac{a+b}{2} = 2\sqrt{ab}\] Solving this equation gives us: \[\frac{a}{\sqrt{ab}} + \frac{b}{\sqrt{ab}} = 4\] Simplifying this equation gives us: \[\sqrt{\frac{a}{b}} + \sqrt{\frac{b}{a}} = 4\] Show more…
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The arithmetic mean between two positive numbers a and b where (a > b) is twice their geometric mean. Then $\frac{\text { a }}{\mathrm{b}}$ (a) $2+\sqrt{3}$ (b) $7+4 \sqrt{3}$ (c) $2-\sqrt{3}$ (d) $7-4 \sqrt{3}$
If $A_{1}, A_{2}$ be two arithmetic means and $G_{1}, G_{2}$ be two geometric means between two positive numbers a and $b$, then $\frac{A_{1}+A_{2}}{G_{1} G_{2}}$ is cqual to (a) $\frac{a}{b}+\frac{b}{a}$ (b) $\frac{1}{a}+\frac{1}{b}$ (c) $\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}$ (d) $\frac{a b}{a+b}$
The arithmetic mean of two numbers $a$ and $b$ is defined as $(a+b) / 2 ;$ the geometric mean of two positive numbers $a$ and $b$ is defined as $\sqrt{a b}$. (a) For two positive numbers, which of the two means is larger? Justify your answer. [Hint: Define $f(x)=(a+x) / 2-\sqrt{a x} \text { for fixed } a .]$ (b) For three positive numbers $a, b, c,$ the arithmetic and geometric mean are $(a+b+c) / 3$ and $\sqrt[3]{a b c}$ respectively. Which of the two means of three numbers is larger? [Hint: Redefine $f(x)$ for fixed $a$ and b.]
Using the Derivative
Optimization and Modeling
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