Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(a \neq 0 \Rightarrow a^{2}+1 / a^{2} \geq 2\). Hint: Consider \((a-1 / a)^{2}\)

Step-by-Step Solution

Answer

If \(a \neq 0\), then \(a^2 + \frac{1}{a^2} \geq 2\).

Step 1: Expand the Given Expression

Consider the expression \[ (a - \frac{1}{a})^2. \]Expanding this expression using the distributive property gives:\[ a^2 - 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^2 = a^2 - 2 + \frac{1}{a^2}. \]

Step 2: Use Non-negativity of Squares

Since \[ (a - \frac{1}{a})^2 \geq 0, \]we know that the expression \[ a^2 - 2 + \frac{1}{a^2} \geq 0. \]

Step 3: Re-arrange to the Desired Form

Rearranging the inequality \[ a^2 - 2 + \frac{1}{a^2} \geq 0, \]we add 2 to both sides to get:\[ a^2 + \frac{1}{a^2} \geq 2. \]

Step 4: Conclusion

We have shown that if \(a eq 0\), then\[ a^2 + \frac{1}{a^2} \geq 2. \]This completes the proof using the square of the difference \((a - \frac{1}{a})^2\).

Key Concepts

Quadratic Expressions

A quadratic expression is any mathematical statement that includes a term with the variable raised to the second power. The general form is usually written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic expressions can appear in numerous forms when solving complex problems.
In the given exercise, we deal with expressions like \(a^2\) and \(\frac{1}{a^2}\), which match the characteristics of quadratic expressions. Understanding these allows us to manipulate and solve inequalities involving them.

• Watch for the second-degree term, as it dictates the quadratic nature of the expression.
• Recognizing the structure helps in expanding and simplifying expressions correctly.

Quadratic expressions are fundamental in various branches of mathematics, notably in solving inequalities, as in this problem.

Expanding Expressions

Expanding expressions involves rewriting a product or a power in an expanded form. This technique is especially useful when dealing with expressions that involve factoring, squaring, or multiplying binomials.
In the exercise, we expanded \((a - \frac{1}{a})^2\) to transform it into a more familiar form:

• Apply the distributive property: it's essential when expanding expressions like \((a - \frac{1}{a})^2 = a^2 - 2\cdot a\cdot \frac{1}{a} + \left(\frac{1}{a}\right)^2\).

Notice how this expansion helps identify and combine like terms, facilitating further solving steps, such as simplifying and rearranging the expression. Recognizing when and how to expand expressions is a crucial skill for solving a wide range of mathematical problems.

Non-negativity

Non-negativity is a critical concept when dealing with inequalities. A non-negative expression is one that is greater than or equal to zero.
In this problem, we used the non-negativity of \((a - \frac{1}{a})^2\), which means it is always \(\geq 0\). This property is fundamental, and understanding it can help us rearrange inequalities and deduce useful relationships.

• Remember: the square of any real number is always non-negative.
• Use this property to establish boundaries or constraints within inequalities; it often leads directly to the solution.

Non-negativity itself is simple but tremendously powerful in mathematical proofs and solutions.

Mathematical Proofs

Mathematical proofs demonstrate the truth of a statement using a systematic series of logical steps. Proofs rely heavily on logical reasoning, known mathematical principles, and properties.
In the exercise, we used the expansion and non-negativity to craft a proof that \(a^2 + \frac{1}{a^2} \geq 2\). Reflect on the following:

• Each step logically follows from the previous one, ensuring that our conclusion is sound.
• Proofs vary in complexity but always aim to bridge given or known information with a new, often insightful result.

Developing an understanding of how to construct and follow a proof is a significant milestone in mastering higher mathematics. Always aim to explain each step clearly, leaving no gaps in logic, just as demonstrated in this exercise.