Problem 102 - College Algebra

Question

Suppose that a function \(f\) is increasing on \((a, b)\) and decreasing on \((b, c) .\) Describe what occurs at \(x=b\) What does the function value \(f(b)\) represent?

Step-by-Step Solution

Answer

At \(x = b\), the function switches from increasing to decreasing, making \(x = b\) a local maximum or 'peak' of the function. The function value at this point, \(f(b)\), is the local maximum value of the function over the interval (a, c). For all x in this interval, the function values \(f(x)\) are less than or equal to \(f(b)\).

Step 1: Understand the Behavior of Increasing and Decreasing Functions

In the interval \((a, b)\), the function is increasing. This means that for any two numbers \(x_1\) and \(x_2\) in this interval, if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). Conversely, in the interval \((b, c)\), the function is decreasing. So, for any two numbers \(x_1\) and \(x_2\) in this interval, if \(x_1 < x_2\), then \(f(x_1) > f(x_2)\).

Step 2: Identify the Turning Point

From step 1, we see that the function changes from increasing to decreasing at \(x = b\). This point, \(x = b\) is known as a 'turning point' or 'peak'. In our case, since the function changes from increasing to decreasing at \(b\), \(x = b\) is a 'local maximum' or 'peak' of the function.

Step 3: Identify the Property of the Function Value at the Turning Point

At \(x = b\), \(f(b)\) represents the local maximum value of the function on the interval \((a, c)\). For all \(x\) in \((a, c)\), the function values \(f(x)\) will be less than or equal to \(f(b)\).

Key Concepts

Increasing Function

An increasing function is one where, as you move from left to right on the graph (from a smaller to a larger x-value), the y-values of the function rise. This indicates that the function is climbing upwards. Mathematically, a function \( f \) is increasing on an interval \((a, b)\) if for any two numbers \( x_1 \) and \( x_2 \) in the interval, whenever \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \).
In simpler terms:

• The function's output (\( f(x) \)) keeps getting larger as you pick numbers (inputs) closer to \( b \) from the left of the interval.
• An increasing function often looks like a hill rising as you travel from left to right.

Understanding when a function is increasing tells us a lot about how the function behaves in those particular sections of the graph. It is like walking up a hill, you start at a lower point and end at a higher point.

Decreasing Function

A decreasing function behaves opposite to an increasing function. As you proceed from left to right (from a smaller to a larger x-value), the function values drop. This means the graph slants downwards. A function \( f \) is said to be decreasing on an interval \((b, c)\) if for any two numbers \( x_1 \) and \( x_2 \) in this interval, whenever \( x_1 < x_2 \), \( f(x_1) > f(x_2) \).
To break it down:

• As you go from \( b \) to \( c \), the outputs (\( f(x) \)) diminish in size.
• Graphically, it resembles descending a slope.

This pattern is typical of a function after it passes its peak, such as sliding down the other side of a hill. This understanding is essential because it helps to identify sections of the graph where the function's value falls as x increases.

Turning Point

A turning point in a function is where its direction changes. In our example, this point is at \( x = b \), where the function transitions from increasing on \((a, b)\) to decreasing on \((b, c)\). A turning point is often a place where a local maximum or minimum occurs.
Key characteristics:

• It represents the peak of a hill or the bottom of a valley on the graph of the function.
• At \( x = b \), the function reaches a local maximum in our case, meaning \( f(b) \) is greater than all nearby values of \( f(x) \) around \( b \).

Turning points are important analysis points because they signify where the function's behavior changes significantly.Visualizing it is akin to reaching the top of a hill and starting your descent, where the function loses altitude beyond this point.