Problem 39 - Calculus Single Variable

Question

The derivative \(f^{\prime}\) of a function \(f\) is given. Determine and classify all local extrema of \(f\). $$ f^{\prime}(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

Function \(f\) has a local maximum at \(x = -1\) and a local minimum at \(x = 1\).

Step 1: Find Critical Points

To find the critical points of the function, set the derivative equal to zero and solve for \(x\): \[ f^{prime}(x) = x^2 - 1 = 0 \] This simplifies to: \[ x^2 = 1 \] Taking the square root of both sides yields the solutions: \[ x =
cpm 1 \] The critical points are therefore \(x = -1\) and \(x = 1\).

Step 2: Determine the Sign of the Derivative

To classify these critical points, examine the sign of \(f^{prime}(x)\) around them. Consider intervals divided by these critical points: - Interval 1: \((-infty, -1)\) - Interval 2: \((-1, 1)\) - Interval 3: \((1, infty)\).Choose a test point from each interval:- Test point for Interval 1: \(x = -2\) - Test point for Interval 2: \(x = 0\) - Test point for Interval 3: \(x = 2\).

Step 3: Evaluate Derivative at Test Points

Calculate \(f^{prime}(x)\) at each test point:- For \(x = -2\): \[ f^{prime}(-2) = (-2)^2 - 1 = 4 - 1 = 3, \] which is positive.- For \(x = 0\): \[ f^{prime}(0) = 0^2 - 1 = -1, \] which is negative.- For \(x = 2\): \[ f^{prime}(2) = (2)^2 - 1 = 4 - 1 = 3, \] which is positive.

Step 4: Classify Critical Points

By examining the sign changes around critical points:- At \(x = -1\): - \(f^{prime}(x)\) changes from positive (Interval 1) to negative (Interval 2). - Therefore, \(f(x)\) has a local maximum at \(x = -1\).- At \(x = 1\): - \(f^{prime}(x)\) changes from negative (Interval 2) to positive (Interval 3). - Therefore, \(f(x)\) has a local minimum at \(x = 1\).

Key Concepts

Derivatives

Derivatives are fundamental in understanding the behavior of functions. They measure how a function changes as its input changes. Essentially, the derivative at a particular point gives the slope of the tangent line to the graph of the function at that point.

For a function like \[ f(x) \], its first derivative, represented as \( f'(x) \), helps us determine where the function is increasing or decreasing:

• If \( f'(x) > 0 \), the function is increasing at that interval.
• If \( f'(x) < 0 \), the function is decreasing at that interval.

When solving for critical points, set the first derivative \( f'(x) \) equal to zero. This step helps find where the function's slope is neither increasing nor decreasing, which often corresponds to local extrema.

Local Extrema

Local extrema are specific points on a graph where the function reaches either a local maximum or a local minimum. These points are crucial in understanding the overall behavior and shape of the function's graph.

At a local maximum:

• The function changes from increasing to decreasing.
• The derivative \( f'(x) \) changes from positive to negative.

At a local minimum:

• The function changes from decreasing to increasing.
• The derivative \( f'(x) \) changes from negative to positive.

In order to find and classify these points, determine the sign changes in \( f'(x) \) around each critical point. This process provides a clear indication of whether you've located a maximum or a minimum.

Sign Analysis

Sign analysis involves studying the behavior of the derivative across different intervals of the function. By analyzing these intervals, we gain important insights into where a function increases or decreases.

To perform sign analysis, follow these steps:

• Divide your number line into intervals separated by the critical points.
• Select a test point within each interval and plug it into \( f'(x) \) to determine the sign of the derivative in that interval.

For instance, for the function with derivative \( f'(x) = x^2 - 1 \), we have the critical points at \( x = -1 \) and \( x = 1 \). By selecting test points like \( x = -2, 0, \) and \( 2 \) for these intervals, you can determine that:

• In the interval \((-\infty, -1)\), \( f'(x) \) is positive, indicating an increasing function.
• In the interval \((-1, 1)\), \( f'(x) \) is negative, indicating a decreasing function.
• In the interval \((1, \infty)\), \( f'(x) \) is positive again.

This sign analysis helps confirm the type of local extrema at each critical point, offering a vital tool to fully grasp the function's behavior.