Problem 23 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x, y, z\) are three real numbers of the same sign then the value of \(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\) lies in the interval (A) \([2, \infty)\) (B) \([3, \infty)\) (C) \((3, \infty)\) (D) \((-\infty, 3)\)

Step-by-Step Solution

Answer

The value lies in the interval \([3, \infty)\).

Step 1: Analyze the Expression

The expression given is \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\). Let's find out if there are any constraints or simplifications based on the problem statement. We know that \(x, y, z\) are of the same sign, which means either all are positive or all are negative.

Step 2: Apply the AM-GM Inequality

We can apply the AM-GM inequality to \(\frac{x}{y}, \frac{y}{z}, \frac{z}{x}\). According to the AM-GM inequality for any non-negative numbers, \[\frac{\frac{x}{y} + \frac{y}{z} + \frac{z}{x}}{3} \geq \sqrt[3]{\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x}}\].The product \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x} = 1\). Thus, \[\frac{x}{y} + \frac{y}{z} + \frac{z}{x} \geq 3.\]

Step 3: Determine if Equality Holds

The equality holds when \(\frac{x}{y} = \frac{y}{z} = \frac{z}{x} = 1\), meaning \(x = y = z\). The minimum value of the expression \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\) is then 3.

Step 4: Conclusion on the Interval

Since \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\) is always greater than or equal to 3 when \(x, y, z\) are of the same sign, the values cannot be less than 3. Thus, the solution falls in the interval \([3, \infty)\).

Key Concepts

Real Numbers

Real numbers encompass all the numbers on the infinite number line. They include rational numbers, such as fractions, and irrational numbers, like the square root of 2. In mathematics, the set of real numbers is critical because they are used to represent continuous quantities. This makes them fundamental in many fields, including calculus and algebra.
When we say that numbers have the "same sign," it implies that they're either all positive or all negative. This uniformity in sign ensures that certain mathematical properties hold true. For instance, when numbers are all positive or all negative, we can apply inequalities, like the AM-GM Inequality, without the additional consideration of sign changes affecting the result. This exercise uses real numbers because it ensures that the arithmetic mean-geometric mean (AM-GM) inequality is applicable fully.

Inequalities

Inequalities are mathematical statements that describe the relative size or order of two values. They use symbols like \(<, >, \leq, \geq\) to convey this relationship. The AM-GM Inequality is a famous inequality used in various mathematical proofs and applications. It states that for any list of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.
In the given problem, we apply the AM-GM Inequality to the expression \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\). The inequality tells us that this sum is always at least 3 when \(x, y, z\) are real numbers of the same sign. This is because the geometric mean of the products of these ratios is 1, given that the product across a cycle like this equals 1 exactly when \(x = y = z\). Thus, the problem's solution leverages the AM-GM Inequality's power to establish a minimum bound on our expression.

Expressions

An expression is a combination of numbers and symbols that represent a mathematical object or relationship. In this exercise, the expression in question is \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\).
Expressions are at the core of algebra and calculus because they allow us to succinctly represent and manipulate mathematical relationships. They can be evaluated to find a numerical value or analyzed to reveal properties like minimum values or behaviors over certain intervals.
In this particular expression, the variables \(x\), \(y\), and \(z\) take on real number values with the condition that they share the same sign. By analyzing this expression, using the AM-GM Inequality helps us determine that the smallest value the expression can take is 3. This shows how expressions can be explored not just to calculate values but to understand the range or behavior of the function they define.