Inequality of Arithmetic and Geometric Means - Example Application

Example Application

Consider the function

for all positive real numbers x, y and z. Suppose we wish to find the minimal value of this function. First we rewrite it a bit:

\begin{align} f(x,y,z) &= 6 \cdot \frac{ \frac{x}{y} + \frac{1}{2} \sqrt{\frac{y}{z}} + \frac{1}{2} \sqrt{\frac{y}{z}} + \frac{1}{3} \sqrt{\frac{z}{x}} + \frac{1}{3} \sqrt{\frac{z}{x}} + \frac{1}{3} \sqrt{\frac{z}{x}} }{6}\\ &=6\cdot\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6} \end{align}

with

Applying the AM–GM inequality for n = 6, we get

\begin{align} f(x,y,z) &\ge 6 \cdot \sqrt{ \frac{x}{y} \cdot \frac{1}{2} \sqrt{\frac{y}{z}} \cdot \frac{1}{2} \sqrt{\frac{y}{z}} \cdot \frac{1}{3} \sqrt{\frac{z}{x}} \cdot \frac{1}{3} \sqrt{\frac{z}{x}} \cdot \frac{1}{3} \sqrt{\frac{z}{x}} }\\ &= 6 \cdot \sqrt{ \frac{1}{2 \cdot 2 \cdot 3 \cdot 3 \cdot 3} \frac{x}{y} \frac{y}{z} \frac{z}{x} }\\ &= 2^{2/3} \cdot 3^{1/2}. \end{align}

Further, we know that the two sides are equal exactly when all the terms of the mean are equal:

All the points (x,y,z) satisfying these conditions lie on a half-line starting at the origin and are given by

Read more about this topic:  Inequality Of Arithmetic And Geometric Means

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