Question

Use the Monotonicity Theorem to prove each statement if \(0\frac{1}{y}\)

Short answer

For (a) and (b), use a strictly increasing function. For (c), use a strictly decreasing function.

Step-by-step solution

Understand Monotonicity Theorem

The Monotonicity Theorem states that if a continuous function is strictly increasing on an interval, then for any two numbers in that interval, if the first number is smaller than the second, the function value of the first number is also smaller than that of the second.

Apply Monotonicity to Part (a)

For the function \(f(x) = x^2\), it is strictly increasing on the interval \((0, \infty)\). Therefore, if \(0 < x < y\), then \(x^2 < y^2\) as \(f(x)\) is strictly increasing.

Apply Monotonicity to Part (b)

Consider the function \(f(x) = \sqrt{x}\). This function is also strictly increasing on \((0, \infty)\). Hence, for \(0 < x < y\), we have \(\sqrt{x} < \sqrt{y}\) because \(f(x) = \sqrt{x}\) is strictly increasing.

Apply Monotonicity to Part (c)

Consider the function \(f(x) = \frac{1}{x}\). This function is strictly decreasing on the interval \((0, \infty)\), meaning that if \(0 < x < y\), then \(\frac{1}{x} > \frac{1}{y}\) since \(f(x)\) is strictly decreasing.

Key concepts

Increasing Function

An increasing function is a type of function where the value of the function goes up as the input value increases. In mathematical terms, if a function \( f(x) \) is considered strictly increasing on an interval, then for any two points \( x \) and \( y \) within that interval where \( x < y \), it follows that \( f(x) < f(y) \). This concept means that as you move from left to right on a graph of the function, the graph trends upward.

• The Monotonicity Theorem applies directly to increasing functions and helps prove inequalities by considering the relative positions of two points on a number line or an interval.
• For example, the function \( f(x) = x^2 \) is strictly increasing on the interval \( (0, \infty) \). If we know that \( 0 < x < y \), we can confidently conclude that \( x^2 < y^2 \) using the theorem.

Decreasing Function

A decreasing function behaves in the opposite manner compared to an increasing function. As the input value increases, the output value decreases. Formally, a function \( f(x) \) is strictly decreasing on an interval if, for any two numbers \( x \) and \( y \) such that \( x < y \), then \( f(x) > f(y) \). This means the function values drop as you move from left to right across the graph.

• In mathematical proofs, the Monotonicity Theorem can be utilized to validate inequalities for decreasing functions.
• For instance, consider the function \( f(x) = \frac{1}{x} \), which is strictly decreasing on \( (0, \infty) \). Thus, given \( 0 < x < y \), one can assert that \( \frac{1}{x} > \frac{1}{y} \).

Continuous Function

A continuous function is defined as a function that has no breaks, jumps, or gaps in its graph. This function characteristic is crucial when discussing the Monotonicity Theorem. The theorem relies on the continuous nature of functions to ensure that they behave consistently over an interval.

• A function is continuous on an interval if, for every point within that interval, small changes in the input result in small changes in the output.
• Both increasing and decreasing functions usually must be continuous for the Monotonicity Theorem to apply effectively. This means if we pick any two points on the continuous curve, there is no sudden change in direction between them.
• For example, the functions \( f(x) = x^2 \) and \( f(x) = \sqrt{x} \) are both continuous on \( (0, \infty) \). Hence, leveraging their continuity, you can apply the Monotonicity Theorem to prove inequalities effectively.