Question

T/F: Functions always switch from increasing to decreasing, or decreasing to increasing, at critical points.

Short answer

False, functions do not always switch increasing/decreasing at critical points.

Step-by-step solution

Understanding Critical Points

A critical point of a function is where its derivative is zero or undefined. At these points, the function can potentially change its increasing or decreasing behavior.

Identify Types of Critical Points

Critical points can be classified as local maxima, local minima, or inflection points. However, an inflection point, which occurs when the concavity of the function changes, does not necessarily involve a change from increasing to decreasing or vice versa.

Examine Increasing and Decreasing Behavior at Critical Points

For a function to switch from increasing to decreasing (or vice versa), the derivative must change sign at a critical point. This is true for local maxima or minima, but at inflection points, the function may continue to increase or decrease without switching the direction.

Conclusion Based on Analysis

Given that inflection points can occur at critical points without a change in increasing or decreasing behavior, it is not true that functions always switch from increasing to decreasing, or vice versa, at critical points.

Key concepts

Increasing and Decreasing Functions

A function is considered to be increasing on an interval when its outputs grow larger as its inputs increase. If you imagine a line drawn on a graph, the line goes up as you move from left to right. Conversely, a function is decreasing when its outputs shrink as the inputs increase, which means the line goes down as we move from left to right.

To determine where a function is increasing or decreasing, we look at the sign of the derivative. The derivative of a function tells us the rate at which the function's output is changing at any given point. Here’s the key:

• If the derivative is positive (\( f'(x) > 0 \)), the function is increasing.
• If the derivative is negative (\( f'(x) < 0 \)), the function is decreasing.

Critical points play a vital role in this analysis because they represent potential locations where the function might change its increasing or decreasing nature. However, as we shall see, not every critical point results in such a change.

Inflection Points

Inflection points are special types of points on the curve of a function where its concavity changes. Concavity tells us the "direction" of the curve – whether it is bending upwards like a cup (\( f''(x) > 0 \), called concave up) or downwards like a frown (\( f''(x) < 0 \), called concave down).

At an inflection point, the second derivative of the function changes sign. However, it is possible for a function to be increasing or decreasing even as it passes through an inflection point. This means that:

• The function could continue increasing on both sides of the inflection point.
• The function could continue decreasing on both sides.
• Or, there might be a switch from increasing to decreasing at an inflection point.

So, while inflection points reflect a change in curvature or "shape" of the graph, they do not necessarily involve a change in the functional direction itself.

Derivative Analysis

Derivative analysis is a technique used to understand the behavior of a function by studying its derivatives. The first derivative represents the slope of the tangent line to the function's graph at any given point. Knowing whether this slope is positive, negative, or zero assists in deducing where the function increases or decreases.

Critical points are found where the first derivative is zero or undefined. These points are examined further to classify them as local maxima, minima, or potential inflection points. However, as already noted, a zero derivative does not ensure a transition from increasing to decreasing behavior.

• A sign change in \( f'(x) \) from positive to negative marks a local maximum.
• A sign change from negative to positive signifies a local minimum.
• If \( f'(x) \) does not change sign, the point is neither a maximum nor a minimum, possibly indicating an inflection point.

Thus, while finding where the derivative is zero is important, it's the sign change analysis that fully reveals the nature of the function's behavior around critical points.