Problem 49 - Calculus Early Transcendentals

Question

Use any method to find the relative extrema of the function \(f .\) $$ f(x)=\left|3 x-x^{2}\right| $$

Step-by-Step Solution

Answer

The relative maximum of \( f(x) \) occurs at \( x = \frac{3}{2} \) with a value of \( \frac{9}{4} \).

Step 1: Determine Critical Points

To find the critical points, we need to find where the derivative is zero or undefined. The function given is \( f(x) = |3x - x^2| \). This is a piecewise function. We need to consider the function without the absolute value: \( g(x) = 3x - x^2 \) and find where \( g(x) = 0 \). Solve \( 3x - x^2 = 0 \) by factoring: \( x(3 - x) = 0 \), giving us \( x = 0 \) and \( x = 3 \).

Step 2: Consider the Absolute Value

Since the function involves an absolute value, we must consider where the expression inside the absolute value changes sign, i.e., \( 3x - x^2 = 0 \). From Step 1, we know these points are \( x = 0 \) and \( x = 3 \). These are the points where the piecewise nature of the function might change.

Step 3: Analyze Function on Intervals

Now consider the intervals determined by the critical points: \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\). On \((0, 3)\), \(3x - x^2\) is positive, so \( f(x) = 3x - x^2 \) and on \((3, \infty)\), it is negative making \( f(x) = x^2 - 3x \).

Step 4: Differentiation and Finding Extrema

For \((0, 3)\), differentiate \( f(x) = 3x - x^2 \) to get \( f'(x) = 3 - 2x \). Set \( f'(x) = 0 \) to find \( x = \frac{3}{2} \). For \((3, \infty)\), differentiate \( f(x) = x^2 - 3x \) to get \( f'(x) = 2x - 3 \). Set \( f'(x) = 0 \) to find \( x = \frac{3}{2} \). Since \( x = \frac{3}{2} \) is in the interval \((0, 3)\), it is a critical point.

Step 5: Evaluate Function at Critical Points

Evaluate \( f(x) \) at the critical points \( x = 0 \), \( x = 3 \), and \( x = \frac{3}{2} \) to determine the nature of the extrema. \( f(0) = 0 \), \( f(3) = 0 \), and \( f\left(\frac{3}{2}\right) = \left|3\left(\frac{3}{2}\right) - \left(\frac{3}{2}\right)^2\right| = \frac{9}{4} \).

Step 6: Conclude Extrema

From Step 5, \( f\left(\frac{3}{2}\right) = \frac{9}{4} \) is greater than \( f(0) = 0 \) and \( f(3) = 0 \), indicating a relative maximum at \( x = \frac{3}{2} \). Consider the behavior differences across intervals to finalize that there are no minimums here, given each side of the interval heads towards the critical point \( x = 0 \) or \( x = 3 \) back to zero.

Key Concepts

Critical Points

Critical points in calculus are locations on the graph of a function where the derivative is either zero or undefined. These points are important because they can indicate potential extrema, such as maxima or minima.
To find critical points, you take the first derivative of the function and set it equal to zero. Another possibility for a critical point is where the derivative does not exist. For the function \( f(x) = |3x - x^2| \), critical points occur where the inside function changes behavior, leading to potential extrema or points of inflection.

• Calculate the derivative, \( f'(x) \).
• Set \( f'(x) = 0 \).
• Identify points where \( f'(x) \) does not exist.

Understanding critical points is key to navigating through functions that are piecewise or have absolute values, since they dictate changes in behavior or slope.

Relative Extrema

Relative extrema refer to the highest or lowest points in a certain region of a graph. These points are known as relative maxima or minima and are critical in graphing and understanding the behavior of functions.
To identify these locations, look for critical points where the function moves from increasing to decreasing (maxima) or from decreasing to increasing (minima).

• Set the first derivative equal to zero to find critical points.
• Test points between critical points to determine intervals of increase or decrease.
• Confirm changes from one behavior to another to ensure valid extrema occur.

Once we grasp the entire behavior of the function, we can easily spot where it reaches its local peaks (maxima) or valleys (minima).

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over different intervals of their domain. They are particularly common in problems involving absolute values, like in our given exercise.
For example, \( f(x) = |3x - x^2| \) means the behavior of \( f(x) \) can vary and must be split into parts to be fully understood.
The critical step is to:

• Identify where each piece of the function is valid by finding where transitions occur, often connected to where expressions inside absolute values or other restrictions come into play.
• Analyze each piece separately, analyzing derivative and behavior.
• Combine findings to get a complete picture of the function's graph.

Absolutely essential for understanding how functions can change dynamically and predicting function behavior at different points.

Differentiation

Differentiation is the process of finding the derivative, or the slope of a function. This is fundamental for identifying how a function changes at any instant, leading to understanding curves, rates, and more.
The derivative, \( f'(x) \), gives insights into:

• Slope and rate of change.
• Critical points for extrema finding.
• Smooth connections or identifying where it doesn't exist.

In the given solution, differentiation is key. We began by differentiating each piece of the function \( g(x) = 3x - x^2 \) which provided necessary insights into the location of critical points by setting \( f'(x)\) to zero. Mastery of differentiation techniques allows you to handle more complex functions with confidence, setting the stage for interpreting calculus-related problems effectively.