Question

Prove that if \(f(x) \geq 0\) and \(f^{\prime}(x) \geq 0\) on \(I\), then \(f^{2}\) is nondecreasing on \(I .\)

Short answer

Since \( f(x) \geq 0 \) and \( f'(x) \geq 0 \), \( f^2(x) \) is non-decreasing on \( I \).

Step-by-step solution

Expression for f^2(x)

To understand how the function \( f^2(x) \) changes, we need to find its derivative. The function \( f^2(x) \) is obtained by squaring \( f(x) \). Therefore, \( f^2(x) = (f(x))^2 \).

Differentiate f^2(x)

Apply the chain rule to differentiate \( f^2(x) \). The derivative is \( \frac{d}{dx}(f^2(x)) = 2f(x)f'(x) \).

Analyze the Sign of the Derivative

Since it's given that \( f(x) \geq 0 \) and \( f'(x) \geq 0 \) for all \( x \) in \( I \), then both \( 2f(x) \) and \( f'(x) \) are non-negative. Thus, their product \( 2f(x)f'(x) \) is also non-negative.

Conclude on Monotonicity

Because the derivative \( 2f(x)f'(x) \) is non-negative for all \( x \in I \), the function \( f^2(x) \) is non-decreasing on the interval \( I \). This fulfills the condition for monotonicity.

Key concepts

Mathematical Derivatives

When we talk about derivatives, we are referring to a tool in calculus that helps us understand how a function changes at any given point. Simply put, a derivative tells you the "rate of change" or the "slope" of a function. This is crucial when examining how functions behave. For a function \( f(x) \), the derivative \( f'(x) \) gives us important information about the steepness or direction of the curve at any point on its graph.
To differentiate compound functions like \( f^2(x) \), which is essentially squaring a function \( f(x) \), we use a rule called the chain rule. The chain rule is like following a recipe that helps compute the derivative of composite functions. According to the chain rule:

• Find the derivative of the outer function (in this case, squaring or \( 2f(x) \)).
• Multiply it by the derivative of the inner function (which is \( f'(x) \)).

The result for \( f^2(x) \) is the derivative \( 2f(x)f'(x) \), providing an understanding of how \( f^2(x) \) behaves as \( x \) changes.

Nondecreasing Functions

A function is called nondecreasing on an interval if, as you move from left to right across the interval, the function's value never goes down. More formally, if \( f(x_1) \leq f(x_2) \) for all \( x_1 \leq x_2 \), then the function is nondecreasing. This does not mean it must always increase; it can stay flat in some regions as well.
In the context of derivative analysis, a nondecreasing function implies that its derivative is non-negative across the interval. For \( f^2(x) \), because its derivative \( 2f(x)f'(x) \) is non-negative (given \( f(x) \geq 0 \) and \( f'(x) \geq 0 \)), \( f^2(x) \) must be nondecreasing.
This means the function does not decrease — its values either increase or stay the same as \( x \) traverses through \( I \). If **both** \( f(x) \) and \( f'(x) \) are positive, it enhances the likelihood that \( f^2(x) \) will strictly increase.

Monotonicity

Monotonicity refers to the behavior of a function and how it either consistently increases or decreases over an interval. For monotonic analysis, the derivative of a function is essentially your best friend. It lets you determine whether the function is increasing, decreasing, or neither on a specified interval.
An essential condition for a function to be non-decreasing (one aspect of monotonicity) is that its derivative should be non-negative for all points in that interval. If the derivative is strictly positive, the function is not just non-decreasing; it is strictly increasing.
In our situation, knowing that \( 2f(x)f'(x) \) is non-negative assures us that \( f^2(x) \) does not decrease across the interval \( I \), thus proving that \( f^2(x) \) satisfies the monotonicity condition of being nondecreasing. Understanding monotonicity through derivative assessment helps define the nature and the behavior of the function systematically across its domain.