Problem 52 - Calculus

Question

True or false? Give an explanation for your answer. The function \(f(x)=x^{2}\) is monotonic on any interval.

Step-by-Step Solution

Answer

False. \(f(x) = x^2\) is not monotonic on intervals that include zero.

Step 1: Understand the Concept of a Monotonic Function

A function is monotonic on an interval if it is either entirely non-increasing or non-decreasing throughout the interval. This means that the function must not switch between increasing and decreasing behaviors.

Step 2: Analyze the Behavior of the Function

The function given is \(f(x) = x^2\). To determine whether it is monotonic, we need to check if it is consistently increasing or decreasing in an interval. Find the derivative of \(f(x)\) to analyze its behavior.

Step 3: Calculate the Derivative of f(x)

The derivative of \(f(x) = x^2\) is \(f'(x) = 2x\). The derivative tells us the slope of the tangent line at any point on the function, revealing whether the function is increasing or decreasing.

Step 4: Determine the Intervals of Monotonicity

Analyze the sign of \(f'(x) = 2x\):- For \(x > 0\), \(f'(x) = 2x > 0\), so \(f(x)\) is increasing.- For \(x < 0\), \(f'(x) = 2x < 0\), so \(f(x)\) is decreasing.- At \(x = 0\), \(f'(x) = 0\), indicating a stationary point.Thus, \(f(x) = x^2\) is not consistently increasing or decreasing over intervals that include both positive and negative numbers.

Step 5: Conclusion

\(f(x) = x^2\) is not monotonic over intervals that include both positive and negative numbers; it changes from decreasing to increasing as \(x\) moves from negative to positive across zero. However, it is monotonic on intervals that do not include zero, such as \((0, \, \infty)\) or \((-\infty, \, 0)\).

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that helps us determine the rate of change of a function. When we take the derivative of a function, we obtain a new function that tells us how the original function behaves. For example:

• The derivative can reveal where a function is increasing, decreasing, or constant.
• It offers insights into the slope of the tangent line at any point on the curve of the function.

For the function given in the exercise, which is \(f(x) = x^2\), the derivative is \(f'(x) = 2x\). This means at any point \(x\), the slope is \(2x\). If \(x\) is positive, the slope is positive, indicating an increasing behavior. Conversely, if \(x\) is negative, the slope is also negative, showing a decreasing trend.

Increasing Function

An increasing function is one that rises as you move along the x-axis from left to right. For an interval where a function is increasing:

• The derivative is positive throughout that interval.
• Every succeeding value of the function is greater than the preceding one.

Considering \(f(x) = x^2\) and its derivative \(f'(x) = 2x\), the function increases when \(x > 0\). This is because the derivative \(2x\) is positive in this region. Thus, on the interval \((0, \, \infty)\), \(f(x)\) consistently increases.

Decreasing Function

A decreasing function, in contrast, is one that falls as you move along the x-axis from left to right. Within a decreasing interval:

• The derivative is negative throughout that interval.
• Each subsequent value of the function is less than the previous one.

For the same function \(f(x) = x^2\), the derivative \(f'(x) = 2x\) indicates a decreasing function when \(x < 0\). Here, the negative values of \(x\) lead the derivative \(2x\) to be negative, confirming the function is decreasing within the interval \((-\infty, \, 0)\).

Interval Analysis

Interval analysis helps to determine the areas where a function's behavior changes. To identify these:

• Check the sign of the derivative across different intervals.
• Determine where the derivative equals zero, which indicates stationary points such as peaks or troughs.

For \(f(x) = x^2\), analyzing the derivative shows:

• On \((-\infty, \, 0)\), \(f(x)\) decreases as \(f'(x) = 2x\) is negative.
• On \((0, \, \infty)\), \(f(x)\) increases with a positive \(f'(x)\).
• At \(x = 0\), the derivative is zero, providing a transition point—a minimum for \(f(x) = x^2\).

By understanding these intervals, we can see that \(f(x) = x^2\) is not monotonic across all real numbers, but it is within specific subsets of them.