Problem 64 - Calculus Single Variable

Question

If \(f\) is increasing on an interval \(I,\) does it follow that \(f^{2}\) is increasing? What if the range of \(f\) is \((0, \infty)\) ?

Step-by-Step Solution

Answer

No, not in general, but yes if the range of \(f\) is \((0, \infty)\).

Step 1: Understanding the Problem

We need to determine whether the function squared, \(f^2\), is also increasing on the interval \(I\) given that \(f\) is increasing on \(I\). Additionally, we will consider the special case where the range of \(f\) is \((0, \infty)\).

Step 2: General Case Analysis

For \(f^2\) to be increasing, \(f^2(x_1) < f^2(x_2)\) must hold for any \(x_1 < x_2\) in \(I\). This can be rewritten as \((f(x_2) - f(x_1))(f(x_2) + f(x_1)) > 0\). If \(f\) is increasing, \(f(x_2) \geq f(x_1)\). However, if \(f(x_1) < 0\) or \(f(x_2) < 0\), \(f(x_2) + f(x_1)\) could be negative, making the product negative or zero, which means \(f^2\) may not be increasing.

Step 3: Special Case with Positive Range

If the range of \(f\) is \((0, \infty)\), then \(f(x) > 0\) for all \(x \in I\). In this case, \(f(x_2) + f(x_1) > 0\) for any \(x_1, x_2 \in I\). Hence, \((f(x_2) - f(x_1))(f(x_2) + f(x_1)) > 0\) if \(f(x_2) > f(x_1)\), confirming that \(f^2\) is increasing when the range is \((0, \infty)\).

Key Concepts

Function Squaring

Function squaring involves taking a function, say \( f(x) \), and creating a new function \( g(x) = f(x)^2 \). It's like applying the operation of multiplication by itself for each output of \( f \). This operation changes how the function behaves, especially in terms of its direction of increase or decrease.
When you square values:

• If \( f(x) \) is a positive number, its squared value will also be positive and larger than the original number.
• If \( f(x) \) is a negative number, its square will still be positive, but smaller in magnitude compared to the original number after squaring.
• Zero remains zero, regardless of squaring.

Understanding the behavior of \( f(x)^2 \) is crucial in comparing it with \( f(x) \). The squaring affects the sign and magnitude, which in turn affects whether the new function is increasing or decreasing. Squaring has the potential to ignore the initial increase in \( f(x) \) when it involves negative values becoming positive.

Range of a Function

The range of a function refers to the set of all possible output values it can produce. When you know that \( f(x) \) has a range of \((0, \infty)\), it implies all outputs are positive.
This characteristic is important:

• If all function values are above zero, their squares will retain the increasing nature of the original function, given \( f \) is increasing.
• This is because the squaring operation will always amplify positive differences.

The concept of a positive range simplifies our analysis when checking if a squared function retains an increasing trend. Specifically, with positive outputs, \( (f(x_2) - f(x_1))(f(x_2) + f(x_1)) > 0 \) straightforwardly remains positive, ensuring \( f^2 \) increases with \( f \).

Positive Interval

The term positive interval denotes an interval where the function outputs remain positive throughout. For increasing functions, especially those with ranges constrained to positive values like \((0, \infty)\), a positive interval ensures that the

• squared function keeps the same increasing behavior as the original.
• In contrast, negative values might obscure the increase by turning them positive upon squaring.

Consequently, when we focus solely on positive intervals, we're assured that the output of \( f(x)^2 \) grows similarly to \( f(x) \), leveraging the consistency of a positive product in \( (f(x_2) - f(x_1))(f(x_2) + f(x_1)) \). This results in unambiguous function growth along these intervals.

Interval Analysis

Interval analysis helps in understanding the behavior of functions across different segments of their domain. It requires examining how the function acts between any two points within the interval.

When analyzing \( f^2(x) \) on an interval:

• We need to ensure that for any two points \( x_1 < x_2 \), \((f(x_2) - f(x_1))(f(x_2) + f(x_1)) > 0 \) holds true.
• This analysis becomes complicated when negative values are involved, as it can affect the sign of the resulting product.

By using interval analysis thoughtfully, especially within positive-only ranges, we can conclude that \( f^2 \) remains increasing just like \( f \), delivering reliable results on growth patterns. This analysis is the key to understanding when a function's squaring will preserve its trend across intervals.