Problem 32 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=5-|x-5| $$

Step-by-Step Solution

Answer

The critical number of the function is at \(x = 5\). The function increases on the interval (-\infty, 5), and decreases on the interval [5, \infty). There is a relative maximum of 5 at \(x = 5\).

Step 1: Establish the Piece-wise Function

We rewrite the function \(f(x) = 5 - |x - 5|\) in piece-wise form. This results in the following: For \(x < 5\), \(f(x) = 5 - (5 - x) = x\) For \(x \geq 5\), \(f(x) = 5 - (x - 5) = 10 - x\)

Step 2: Find the Derivatives and Critical Numbers

Depending on the interval of x, the derivative of \(f(x)\) is: For \(x < 5\), \(f'(x) = 1\) For \(x \geq 5\), \(f'(x) = -1\) The derivative of the function is not defined at \(x = 5\), therefore, \(x = 5\) is the critical number.

Step 3: Determine the Intervals of Increase and Decrease

To find out where the function is increasing or decreasing, we check the signs of \(f'(x)\): For \(f'(x) = 1\), (\(x < 5\)), the function is increasing, so \(f(x)\) is increasing on the interval \((- \infty, 5)\). For \(f'(x) = -1\), (\(x \geq 5\)), the function is decreasing, so \(f(x)\) is decreasing on the interval \([5, \infty)\).

Step 4: Find the Relative Extrema

From the above step, we know the function is increasing on \((- \infty, 5)\) and decreasing on \([5, \infty)\). This means the critical number \(x = 5\) is where the function changes from increasing to decreasing, so it is a relative maximum there. To find out the actual value, we substitute \(x = 5\) into the original function: \(f(5) = 5 - |5 - 5| = 5\). So, the function has a maximum of 5 at \(x = 5\).

Key Concepts

Piecewise Functions

Piecewise functions are a type of function that is defined by different expressions, depending on the input value of the variable. They are particularly useful when a function behaves differently over different intervals.
\(\ \)
In our exercise, the function given is \(f(x) = 5 - |x - 5|\). By considering the absolute value part, we need to split this function into two cases: when \(x < 5\) and \(x \geq 5\).
\(\ \)
In other words, piecewise functions enable us to break down complex scenarios into simpler parts, thus allowing us to analyze each part independently. In this exercise, for

• \(x < 5\), the function becomes \(f(x) = x\),
• \(x \geq 5\), it becomes \(f(x) = 10 - x\).

This split helps us understand the behavior of the function over different parts of the number line.

Derivative Analysis

Derivative analysis is a technique used in calculus to determine the rate at which a function is changing. By calculating the derivative, we can understand the behavior of the function, such as identifying turning points and analyzing how the function increases or decreases.
\(\ \)
For a piecewise function like the one given, we calculate the derivative over each piece independently.
\(\ \)
For

• \(x < 5\), the derivative \(f'(x) = 1\)
• \(x \geq 5\), the derivative \(f'(x) = -1\)

These derivatives tell us how the function behaves:

• A derivative of \(1\) means that the function is increasing at a constant rate as \(x\) approaches \(5\) from the left.
• A derivative of \(-1\) means that the function decreases at a constant rate once \(x\) is \(5\) or greater.

It is also important to note that the derivative does not exist at the point where the behavior changes, specifically at \(x = 5\) in this case.

Relative Extrema

Relative extrema are points in the function where the function reaches a local maximum or minimum. These are critical points where you often observe changes in the direction of the function graph.
\(\ \)
In our example, the critical point is found by identifying where \(f'(x)\) does not exist: at \(x = 5\). Here, the function moves from increasing to decreasing:

• In the interval \((-\infty, 5)\), the function increases.
• In the interval \([5, \infty)\), the function decreases.

This transition indicates that the point \(x = 5\) is a relative maximum.
\(\ \)
To find the value of the function at this point, substitute \(x = 5\) into the original function to get \(f(5) = 5\). Therefore, the relative maximum is \(5\).

Increasing and Decreasing Intervals

Increasing and decreasing intervals tell us over which parts of the domain a function is rising or falling. This concept is vital in understanding the overall shape and behavior of the function.
\(\ \)
Based on the derivative analysis:

• The function \(f(x)\) is increasing on the interval \((-\infty, 5)\) because the derivative \(f'(x) = 1\) is positive in this region.
• Conversely, \(f(x)\) is decreasing on \([5, \infty)\) as \(f'(x) = -1\) in this region.

These intervals illustrate where the function graph goes up and where it goes down.
\(\ \)
Understanding these intervals aids in sketching the shape of the function graph, guiding us in finding any local peaks or troughs, which are crucial in identifying relative extrema.