Problem 51 - Calculus

Question

True or false? Give an explanation for your answer. The function \(f(x)=x^{3}\) is monotonic on any interval.

Step-by-Step Solution

Answer

True. The function is non-decreasing on any interval.

Step 1: Understanding Monotonicity

A function is said to be monotonic on an interval if it is either entirely non-increasing or non-decreasing throughout that interval. Mathematically, if a function is increasing, then for any two points \(x_1 < x_2\), we have \(f(x_1) \leq f(x_2)\). Conversely, if it is decreasing, then \(f(x_1) \geq f(x_2)\).

Step 2: Analyzing the Function

The function given is \(f(x) = x^3\). To determine if it is monotonic, we examine the derivative, \(f'(x)\). The derivative \(f'(x) = 3x^2\) is critical in determining monotonicity.

Step 3: Calculating the Derivative

Evaluate the derivative \(f'(x) = 3x^2\). Since \(3x^2\) is non-negative for all real numbers \(x\), it means that \(f(x)\) is non-decreasing for all values of \(x\).

Step 4: Conclusion on Monotonicity

Since the derivative \(f'(x) = 3x^2\) is greater than or equal to zero for all values of \(x\), the function \(f(x) = x^3\) is indeed monotonic (specifically, non-decreasing) on any interval in the real number domain. Thus, the function is considered monotonic on any interval.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus, representing the rate at which a function is changing at any given point. This concept is intimately linked with the idea of the instantaneous rate of change, similar to how velocity indicates the speed of an object at a specific moment. A derivative can be denoted as \( f'(x) \) or \( \frac{df}{dx} \), indicating it is the derivative of function \( f \) with respect to variable \( x \).

• If \( f'(x) > 0 \) over an interval, the function is strictly increasing on that interval.
• If \( f'(x) < 0 \), it is strictly decreasing.
• If \( f'(x) = 0 \), it suggests where the function may have a horizontal tangent line, such as at a local maximum or minimum.

The derivative of the function \( f(x) = x^3 \) is calculated as \( f'(x) = 3x^2 \). This derivative highlights a constant positive rate of change (or zero rate at \( x = 0 \)), indicating that the function is always increasing.

Increasing Function

An increasing function is one where the value of the function grows as you move from left to right along the x-axis. More formally, a function \( f(x) \) is increasing on an interval if, whenever \( x_1 \leq x_2 \), it follows that \( f(x_1) \leq f(x_2) \).

• In particular, if \( f '(x) > 0 \) for all \( x \) in an open interval, \( f(x) \) is strictly increasing.
• For example, the derivative \( f'(x) = 3x^2 \) for the function \( x^3 \) is always non-negative for any \( x \), indicating any segment of this function does not decrease.

Thus, the graph of \( x^3 \) will continue to rise as \( x \) increases.

Non-decreasing Function

A non-decreasing function is one that does not go down as you move across its domain. This means that, to move from one point to another in the function's domain, the function value either stays the same or increases. This is expressed mathematically as: \( f(x_1) \leq f(x_2) \) whenever \( x_1 \leq x_2 \).

• The term 'non-decreasing' is slightly broader than 'increasing' since it includes the possibility where parts of the function are flat (i.e., \( f'(x) = 0 \)).
• The function \( x^3 \) fits well into this category as its derivative, \( 3x^2 \), never becomes negative, suggesting its slope is either flat or positive.

Hence, in any interval you inspect, \( x^3 \) continues to either stay level or climb, making it a classic example of a non-decreasing function.