Problem 101 - College Algebra Essentials

Question

Suppose that a function \(f\) whose graph contains no breaks or gaps on \((a, c)\) is increasing on \((a, b),\) decreasing on \((b, c)\) and defined at \(b\). Describe what occurs at \(x=b\). What does the function value \(f(b)\) represent?

Step-by-Step Solution

Answer

The function \(f\) reaches its maximum value at the point \(x=b\). So, \(f(b)\) represents the maximum value of the function on the interval \((a, c)\), making it a relative maximum.

Step 1: Identify Function Behavior

Given the function \(f\) is increasing on the interval \((a, b)\) and decreasing on the interval \((b, c)\). By definition, a function is said to be increasing on an interval if for any two numbers \(x_1\) and \(x_2\) in the interval, \(x_1 < x_2 \Rightarrow f(x_1) < f(x_2)\). Similarly, a function is decreasing on an interval if for any two numbers \(x_1\) and \(x_2\) in the interval, \(x_1 < x_2 \Rightarrow f(x_1) > f(x_2)\).

Step 2: Recognize the Transition Point

Notice that the transition from increasing to decreasing occurs at \(x = b\). This means that all values of the function to the left of \(b\) (on the interval \((a, b)\)) are increasing, and all values to the right of \(b\) (on the interval \((b, c)\)) are decreasing. This is a sign that the function reaches a peak or maximum at \(x = b\).

Step 3: Interpret the Function Value

Therefore, the function value \(f(b)\) represents the maximum value of the function on the interval \((a, c)\). In other words, for all \(x\) in \((a, c)\), the largest value that \(f(x)\) obtains is \(f(b)\).

Step 4: Conclusion

So, based on the described behavior of the function \(f\), it's feasible to say that at \(x=b\), the function reaches its peak or maximum value, which is \(f(b)\). This point is considered a relative maximum.