Problem 83 - Precalculus with Limits

Question

THINK ABOUT IT Sketch the graph of a function for which \(f'(x) < 0\) for \(x < 1\), \(f'(x) \geq 0\) for \(x > 1\), and \(f'(1) = 0\).

Step-by-Step Solution

Answer

The graph should start decreasing for \(x<1\), then at \(x=1\) the slope should be flat before finally either increasing or staying flat for \(x>1\).

Step 1: Identify the intervals

The task gave three conditions for different intervals of x, which divide the real numbers into three regions: \(x < 1\), \(x = 1\), and \(x > 1\).

Step 2: Sketch the graph based on the function's behavior

From step 1, we know that the function decreases on the interval \(x<1\), so the graph should be drawn downwards as x approaches 1 from the left. At \(x=1\), the derivative is equal to 0, which means the graph should have a horizontal tangent at \(x=1\). Lastly, when \(x>1\), the function does not decrease, so the graph should be drawn upwards or flat as x moves away from 1 to the right.

Step 3: Finalize the graph

Combine parts from step 2 to obtain complete graph. The graph starts downslope for \(x<1\), changes to a flat tangent at \(x=1\), and eventually goes upslope or flat for \(x>1\).

Key Concepts

Derivatives

Derivatives are a key concept in calculus, mainly used to determine how a function changes at any given point. They give us the slope of the function's graph at a particular point, which is the rate of change of the function.
For a function \(f(x)\), the derivative \(f'(x)\) tells us:

• If \(f'(x) > 0\): the function is increasing at that point.
• If \(f'(x) < 0\): the function is decreasing at that point.
• If \(f'(x) = 0\): there's a possibility of turning points such as maximum, minimum, or a horizontal inflection point.

In the provided exercise, the derivative is negative for \(x < 1\), zero at \(x = 1\), and non-negative for \(x > 1\). This information guides us on how to sketch the graph, showing decreases, constant slopes, or increases.

Graph Sketching

Graph sketching involves drawing a rough graph of a function by using crucial information like derivatives, important points, and behavior trends. It allows us to represent the function visually and understand its behavior without needing precise plotting.
In this context, recognizing the behavior of the function around \(x = 1\) determines how we sketch the curve:

• For \(x < 1\), where \(f'(x) < 0\), start the graph by depicting it as decreasing.
• At \(x = 1\), since \(f'(x) = 0\), represent the graph with a horizontal tangent. This usually indicates a critical point such as a local maximum or minimum or an inflection point.
• For \(x > 1\), \(f'(x) \ge 0\), show the function as increasing or remaining flat.

These steps together help in determining the shape of the overall graph, providing insights into the function's behavior.

Critical Points

Critical points occur when the derivative of a function is zero or undefined. They are significant because they represent places where the function might have a local maxima or minima or even a point of inflection. In simpler terms, critical points are where the behavior of the function likely changes.
In the exercise, \(x = 1\) is a critical point because \(f'(1) = 0\). Here's what this indicates:

• The function can have a change in increasing to decreasing (or vice-versa).
• This point may be where the graph is flat (horizontal tangent), indicating a possible local minimum or maximum.
• Further analysis or information about the second derivative might be needed to determine the specific nature (minimum, maximum, or inflection).

Recognizing critical points is essential in sketching graphs and understanding the overall architecture of functions