Problem 117 - Precalculus 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

At \(x = b\), the function \(f\) reaches a local maximum. The function value \(f(b)\) represents this local maximum value.

Step 1: Understand the nature of the function

The graph of a continuous function f displays no gaps or breaks on the interval (a,c). It's increasing on (a,b) which means as x increases from a to b, f(x) increases. Conversely, it's decreasing on (b,c), as x increases from b to c, f(x) decreases.

Step 2: Analyze the behavior of function at x = b

As x approaches b from the left, the function values are increasing. Once we pass b and let x increase, the function values are decreasing. This means that the function value at x = b is greater than the function values at any other x within the interval (a, c). Hence, the function value at x = b is a maximum value in the interval (a, c).

Step 3: Conclusion

At x = b, the function f reaches its maximum value over the interval (a, c). More specifically, this is a local maximum. The function value f(b) represents this maximum value.

Key Concepts

Continuous Function

A continuous function, in simple terms, is a function that you can draw on paper without lifting your pencil. This means there's a smooth, uninterrupted path along which the function travels. Let’s consider a specific interval, say \(a, c\). If a function is continuous on this interval, it doesn't break, jump, or have holes anywhere within it.
Continuous functions have some important properties:

• They remain defined at every point within the interval; thus, never "missing" a point in their domain.
• For every small segment along the x-axis, there exists a corresponding value on the curve without abrupt changes.

Understanding continuity helps determine how functions behave, especially where they might peak (reach a maximum) or dip (reach a minimum). Without breaks, like having both increasing and decreasing parts together, they can form peaks and troughs.

Increasing Function

An increasing function is one where, as you move from left to right on your graph (or as your x-values grow), your y-values climb. This means visually, the line of the function slopes upwards. This is how an increasing function behaves:

• The function value \(f(x)\) for any two points \(x_1\) and \(x_2\), where \(x_1 < x_2\), satisfies \(f(x_1) < f(x_2)\).
• Its graph showcases a consistent upward trend over the specified interval.

Availability of increasing sections in a function, as seen in \(a, b\) of our interval, suggests ascending behavior leading to possible maxima. This behavior ceases earlier than any maximum point, ensuring a peak is reached somewhere in or at the end of the interval.

Decreasing Function

A decreasing function is a function where, as x increases, the output \(f(x)\) decreases. In plain terms, it's like a downhill slope on a graph. When you observe this behavior:

• For any two points \(x_1\) and \(x_2\) with \(x_1 < x_2\), the function satisfies \(f(x_1) > f(x_2)\).
• The line travels downward, indicating how values fall as the independent variable x progresses.

On intervals like \(b, c\), this describes how the function declines after peaking at the local maximum. Recognizing these decreasing behaviors is pleasing for understanding where functions start to "dip." Each interval offers clues to a function's highest or lowest points, often shown where the function shifts from increasing to decreasing behavior.