Problem 76 - Thomas Calculus

Question

Suppose the derivative of the function \(y=f(x)\) is $$ y^{\prime}=(x-1)^{2}(x-2)(x-4) $$ At what points, if any, does the graph of \(f\) have a local minimum, local maximum, or point of inflection?

Step-by-Step Solution

Answer

Local minimum at \(x=2\), local maximum at \(x=4\), point of inflection at \(x=1\).

Step 1: Determine Critical Points

Critical points occur where the derivative changes sign. Set the derivative equal to zero: \[ (x-1)^2(x-2)(x-4) = 0 \]Solving this gives us critical points at \(x = 1\), \(x = 2\), and \(x = 4\).

Step 2: Analyze the Behavior of the Derivative

Determine the sign of the derivative in the intervals created by the critical points. Use test points in each interval:- For \(x < 1\) (e.g., \(x = 0\)):\[ ((0-1)^2(0-2)(0-4)) > 0 \]- For \(1 < x < 2\) (e.g., \(x = 1.5\)):\[ ((1.5-1)^2(1.5-2)(1.5-4)) < 0 \]- For \(2 < x < 4\) (e.g., \(x = 3\)):\[ ((3-1)^2(3-2)(3-4)) > 0 \]- For \(x > 4\) (e.g., \(x = 5\)):\[ ((5-1)^2(5-2)(5-4)) < 0 \]

Step 3: Determine Type of Critical Points

Using the sign changes from Step 2:- At \(x = 1\): No sign change from negative to negative.\(x = 1\) is not a local extremum.- At \(x = 2\): The sign changes from negative to positive, indicating a local minimum.- At \(x = 4\): The sign changes from positive to negative, indicating a local maximum.

Step 4: Identify Points of Inflection

A point of inflection occurs where the concavity changes, which is typically where the second derivative changes sign. We observe that the derivative itself changes sign at \(x=1\), but also the behavior in the intervals around \(x=1\) indicates a possible point of inflection.\(x = 1\) is a point of inflection as it changes the concavity from up to down.

Key Concepts

Derivative Analysis

Derivative analysis is crucial in understanding how a function behaves. By analyzing derivatives, we can identify critical points where the function's slope is zero or undefined. This helps us determine points of interest on a graph, such as local minima, maxima, or points of inflection.
To find these critical points, we examine where the first derivative of a function equals zero or doesn't exist. For instance, given the derivative \[y^{\prime}=(x-1)^2(x-2)(x-4)\], set this equal to zero to find where potential changes in direction occur. Solving yields the critical points at \(x = 1\), \(x = 2\), and \(x = 4\).
After identifying critical points, the next step is to test intervals between them to see where the derivative changes sign. This involves picking test values from each interval and substituting them into the derivative. If the sign of the derivative changes, it indicates a potential point of interest, such as a local minimum, maximum, or inflection point.

Local Minimum and Maximum

Local minima and maxima are key features of a function's graph, representing the lowest or highest points in a particular section. To determine these points:

• Identify critical points where the derivative is zero or undefined.
• Analyze the behavior of the derivative around these points using test values in the intervals.

For example, if the derivative changes from negative to positive at a certain critical point, it indicates a local minimum. In our function, this occurs at \(x = 2\). Conversely, a change from positive to negative indicates a local maximum, as seen at \(x = 4\).
Thus,

• A local minimum means the function has a "valley," where the function goes from decreasing to increasing.
• A local maximum represents a "peak," where the function transitions from increasing to decreasing.

This step-by-step analysis helps us map out the entire behavior of the function across different intervals.

Points of Inflection

Points of inflection are special points where the function's graph changes concavity. This means the curve switches from being concave up (like a cup) to concave down (like a cap), or vice versa.
To identify points of inflection, we often need to compute and analyze the second derivative, but sometimes the behavior of the first derivative can also provide clues. In the case of the derivative \(y^{\prime}=(x-1)^2(x-2)(x-4)\), the critical point \(x = 1\) is a point of inflection because the derivative changes its behavior subtly around this point.
Although the sign of the derivative at \(x=1\) does not change in a traditional sense (negative to negative), the nature of the curve shifts, indicating a change in concavity. It's key to understand that an inflection point is where the graph's "bend" alters, which is often accompanied by a visual transition in the function's slope.
Recognizing points of inflection helps in predicting the graph's behavior, ensuring a deeper understanding when sketching or analyzing the function.