Problem 14 - Discrete Mathematics with Applications

Question

Prove each directly. The arithmetic mean \(\frac{a+b}{2}\) of any two nonnegative real numbers \(a\) and \(b\) is greater than or equal to their geometric mean \(\sqrt{a b}\) I Hint: consider \((\sqrt{a}-\sqrt{b})^{2} \geq 0.1\)

Step-by-Step Solution

Answer

To prove that the arithmetic mean \(\frac{a+b}{2}\) of any two nonnegative real numbers \(a\) and \(b\) is greater than or equal to their geometric mean \(\sqrt{ab}\), we first consider the expression \((\sqrt{a}-\sqrt{b})^2 \geq 0\). Expanding this expression, we get \(a - 2\sqrt{ab}+ b\). Using the inequality, we have \(a + b \geq 2\sqrt{ab}\). Dividing both sides by 2, we arrive at the desired inequality \(\frac{a+b}{2} \geq \sqrt{ab}\).

Step 1: Expand the hint expression.

Since we know that \((\sqrt{a}-\sqrt{b})^2 \geq 0\), let's expand this expression: \[ (\sqrt{a}-\sqrt{b})^2 = a - 2\sqrt{ab}+ b \] Step 2: Apply the inequality

Step 2: Apply the inequality to the expanded expression.

Since we have the inequality \((\sqrt{a}-\sqrt{b})^2 \geq 0\), let's apply it to the expanded expression: \[ a - 2\sqrt{ab}+ b \geq 0 \] Step 3: Rearrange inequality

Step 3: Rearrange the inequality to isolate terms.

Next, let's rearrange the inequality to get the arithmetic and geometric means on either side: \[ a + b \geq 2\sqrt{ab} \] Step 4: Divide both sides by 2

Step 4: Divide both sides by 2.

To get the desired inequality, divide both sides of the inequality by 2: \[ \frac{a+b}{2} \geq \sqrt{ab} \] This proves that the arithmetic mean of any two nonnegative real numbers is greater than or equal to their geometric mean.

Key Concepts

Inequality Proof

Understanding how to prove an inequality is a foundational mathematical skill. Inequalities tell us about the relative size of two expressions. To prove an inequality means showing that it holds true for the specified range of variables involved.

In our case, we are proving an inequality involving the arithmetic and geometric means of two nonnegative real numbers. The proof uses a clever approach by examining the square of the difference between the square roots of these numbers. Since the square of any real number is nonnegative, we start with the premise \( (\sqrt{a} - \sqrt{b})^2 \geq 0 \). Expanding this expression and manipulating it carefully allows us to demonstrate the desired relationship between the arithmetic and geometric means.

Arithmetic Mean

The arithmetic mean, often referred to as the average, is a measure of central tendency. It is calculated by taking the sum of a set of values and dividing it by the count of the values.

For example, the arithmetic mean of two nonnegative real numbers \(a\) and \(b\) is expressed as \(\frac{a+b}{2}\). It represents a value that is equidistant from \(a\) and \(b\) on the number line. The concept of an average is intuitive and is used extensively in various fields, including statistics, finance, and everyday life. When we talk about averages in a general sense, we are typically referring to the arithmetic mean.

Geometric Mean

In contrast to the arithmetic mean, the geometric mean is a measure that indicates the central tendency of a set of numbers by using the product of their values. Specifically, the geometric mean of two nonnegative real numbers \(a\) and \(b\) is the square root of their product, denoted \(\sqrt{ab}\).

It can be thought of as the 'average' that represents the numbers in a multiplicative sense and is particularly useful when dealing with rates of change or percentages. Geometric mean finds applications in fields like finance, especially in the context of growth rates and returns.

Inequality of Arithmetic and Geometric Means

The inequality of arithmetic and geometric means, also known as the AM-GM inequality, states that for any nonnegative real numbers, the arithmetic mean is always greater than or equal to the geometric mean. This can be written as \(\frac{a+b}{2} \geq \sqrt{ab}\).

The inequality is strict (meaning the mean is greater) unless the two numbers are equal, in which case the two means are the same. Our exercise demonstrates this inequality using a sequence of algebraic steps, starting with a squared expression, leading to the final inequality which embodies a profound mathematical relationship that has powerful implications in various scientific and mathematical contexts.