Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

The first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) a local minimum and \((b)\) a local maximum. \(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)(x-4)\)

Step-by-Step Solution

Answer

Local minima at \(x = 2\) and \(x = 4\); no local maxima.

Step 1: Identify Critical Points

To find the critical points of a function where you might have local minima or maxima, we set the first derivative equal to zero. The given derivative is \( f^{\prime}(x) = (x-1)^2(x-2)^2(x-3)(x-4) \). Setting \( f^{\prime}(x) = 0 \) gives us the critical points at \( x = 1, 2, 3, 4 \).

Step 2: Determine the Sign Changes

Analyze the sign changes of \( f^{\prime}(x) \) around each critical point to determine if they correspond to a minimum or maximum. We use test points in intervals divided by the critical points to assess whether \( f^{\prime}(x) \) changes sign.

Step 3: Test Intervals Around Critical Points

Choose test points in intervals \((-fty, 1), (1, 2), (2, 3), (3, 4), (4, fty)\). Assess the sign changes:- For \( x < 1 \), choose \( x = 0 \), \( f^{\prime}(x) > 0 \).- For \( 1 < x < 2 \), choose \( x = 1.5 \), \( f^{\prime}(x) < 0 \).- For \( 2 < x < 3 \), choose \( x = 2.5 \), \( f^{\prime}(x) > 0 \).- For \( 3 < x < 4 \), choose \( x = 3.5 \), \( f^{\prime}(x) < 0 \).- For \( x > 4 \), choose \( x = 5 \), \( f^{\prime}(x) > 0 \).

Step 4: Determine Local Minima and Maxima

If \( f^{\prime}(x) \) changes from positive to negative, then \( f \) has a local maximum at that point. Conversely, if \( f^{\prime}(x) \) changes from negative to positive, \( f \) has a local minimum:- At \( x = 1 \) and \( x = 3 \), \( f^{\prime}(x) \) does not change sign (remains zero), so neither a minimum nor a maximum.- At \( x = 2 \), \( f^{\prime}(x) \) changes from negative to positive, so a local minimum.- At \( x = 4 \), \( f^{\prime}(x) \) changes from negative to positive, so a local minimum.

Key Concepts

Critical Points

Critical points are vital locations on a function's graph where significant changes in behavior, such as local maxima or minima, might occur. These points can be found by setting the first derivative of the function to zero. This means you are looking for where the slope of the tangent (derivative) is zero, which corresponds to the critical points. For the given derivative \(f^{\prime}(x) = (x-1)^2(x-2)^2(x-3)(x-4)\), setting it equal to zero yields the critical points \(x = 1, 2, 3, 4\). These are the \(x\)-values where the function might have a peak or a valley. Remember, finding critical points is just the start of understanding where local maximums or minimums may exist, not the conclusion about their nature.

First Derivative Test

The First Derivative Test is a useful method to determine if a function has a local minimum or maximum at its critical points. It involves analyzing the sign of the first derivative before and after each critical point to see how the slope changes.

This method is straightforward:

• If the derivative changes from positive to negative, the critical point is a local maximum.
• If the derivative changes from negative to positive, the critical point is a local minimum.
• If there is no sign change (the sign remains the same), the point is neither a maximum nor minimum, and might be a point of inflection.

This test helps you determine the nature of each critical point (whether it's a peak, trough, or neither) by observing these sign changes.

Derivative Analysis

Derivative analysis helps you to interpret and understand the behavior of a function between and at critical points. Here, you analyze the sign of the derivative around each critical point using test intervals.

For example, analyze \(f^{\prime}(x)\) at intervals around critical points such as \((-fty, 1), (1, 2), (2, 3), (3, 4), (4, fty)\). By choosing test points in these intervals (like \(x = 0, 1.5, 2.5, 3.5,\) and \(5\)), we assessed whether \(f^{\prime}(x)\) is positive or negative in these regions.

This perspective provides a clear understanding on how the function behaves, indicating increases and decreases, helping to visually map out where the local extrema occur. Remember, this observation is crucial for confirming the nature of critical points.

Local Minimum and Maximum

A local minimum or maximum occurs at a particular point when the function stops decreasing and starts increasing, or vice versa, around that point. By using the First Derivative Test and analyzing derivative sign changes, we can determine these points effectively.

In the provided example, after applying the test around the critical points, we found:

• At \(x = 2\), \(f^{\prime}(x)\) changes from negative to positive, indicating a local minimum as the function switches from decreasing to increasing.
• At \(x = 4\), a similar change from negative to positive also results in a local minimum.

It's essential to note that in this case, neither \(x = 1\) nor \(x = 3\) are local maxima or minima, as \(f^{\prime}(x)\) remains zero and does not change sign at these points. Recognizing the character of these critical points helps in accurately defining the function's local behavior.