Problem 62 - Thomas Calculus

Question

The graph of \(f^{\prime}\) is given. Assume that \(f\) is continuous and determine the \(x\) -values corresponding to local minima and local maxima.

Step-by-Step Solution

Answer

Local minima occur where \(f'\) changes from negative to positive, and local maxima occur where \(f'\) changes from positive to negative.

Step 1: Identify Critical Points

To find where local minima and maxima occur, we need to identify the critical points of the function. Critical points occur where the derivative \(f'\) is zero or undefined. Look at the graph of \(f'\) and find all \(x\)-values where the graph crosses the \(x\)-axis. These \(x\)-values correspond to critical points.

Step 2: Analyze Sign Changes

To determine whether each critical point is a minimum or maximum, analyze the sign changes of \(f'\) around each critical point. If \(f'\) changes from positive to negative, \(f\) has a local maximum. If \(f'\) changes from negative to positive, \(f\) has a local minimum. Check these sign changes for each critical point identified in Step 1.

Step 3: Determine Local Minima and Maxima

Based on the sign changes of \(f'\) found in Step 2, list the \(x\)-values where \(f\) has a local minimum and where \(f\) has a local maximum. For each critical point, assign it to either the local minima or maxima category based on the direction of the sign change.

Key Concepts

Local Minima

A local minimum is a point on a function where the function value is lower than all nearby points. It's like a small dip in the graph. To find where these points occur, we first identify the critical points of the function by examining the derivative, denoted as \(f'\). Critical points are those values of \(x\) where \(f'(x)\) equals zero or is undefined. Once we find these critical points, we examine the sign changes of the derivative around them.

• If \(f'\) changes from negative to positive as we pass through a critical point, then the original function \(f\) has a local minimum at that point.

This means the graph of \(f\) makes a U-shape at the critical point, rising on both sides. Understanding where local minima occur helps in graphing the function and recognizing its shape.

Local Maxima

Local maxima are just as important to understand as local minima. These are points where the function reaches a higher value than any nearby points, like the peak of a hill. To find local maxima, we return to our critical points found from the derivative \(f'\).

• At a critical point where \(f'\) shifts from positive to negative, \(f\) has a local maximum.

This change in sign indicates that the function is rising before the critical point and falling after it, which forms an upside-down U-shape. Identifying local maxima helps in optimizing functions and understanding their behavior.

Derivative Analysis

Derivative analysis is a crucial step for understanding the behavior of a function. It involves looking at how the derivative \(f'\) behaves, which tells us about the slope of the function \(f\). Here are key points for derivative analysis:

• Critical points occur where \(f'(x) = 0\) or is undefined.
• The sign of \(f'\) indicates whether \(f\) is increasing or decreasing.

If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. By analyzing the derivative, you can determine the shape and turning points of the function. It's like reading the instructions for how the graph moves up and down.

Sign Changes

Sign changes in the derivative are essential for determining local extrema. They give us insight into whether a critical point is a minimum or a maximum. When analyzing the graph of \(f'\):

• If \(f'\) changes from positive to negative, it indicates a local maximum.
• If \(f'\) changes from negative to positive, it indicates a local minimum.

These sign changes occur at critical points and are crucial for categorizing them. By understanding how \(f'\) changes sign, you essentially track the turning points of the graph of \(f\). This method helps predict where the peaks and valleys of the graph occur, aiding in plotting and function optimization.