Question

Fill in the blanks: The critical points of a function \(f\) are found where \(f^{\prime}(x)\) is equal to _______ or where \(f^{\prime}(x)\) is ___________.

Short answer

zero; undefined

Step-by-step solution

Understanding Critical Points

Critical points of a function are the values of \(x\) where the function's first derivative, \(f'(x)\), is either zero or undefined. These points are important because they help identify where the function might have local maxima, minima, or other significant changes in its behavior.

Setting the First Derivative to Zero

One way to find critical points is by setting the first derivative of the function to zero: \(f'(x) = 0\). Solving this equation will give the \(x\)-values where the function has either horizontal tangent lines, potentially indicating local maxima or minima.

Checking for Undefined Derivative

The derivative \(f'(x)\) might also be undefined at certain points. This can happen where there are cusps, vertical tangent lines, or discontinuities in the graph of the function. These points are also considered critical points and need to be identified separately.

Key concepts

First Derivative

The first derivative of a function, often denoted as \(f'(x)\), is a crucial tool in calculus. It represents the rate at which the function's value changes with respect to changes in \(x\). In simpler terms, it tells us how fast or slow a function is growing or shrinking at any given point. Just like the speedometer in a car shows how fast a car is moving, the derivative tells us how fast the function's value is changing.

• A positive first derivative indicates that the function is increasing in that region.
• A negative first derivative points to a decreasing function.
• When the first derivative is zero, the function might have a constant value at that point.

Checking where the first derivative equals zero or is undefined helps in finding critical points.

Local Maxima

Local maxima occur at points where a function reaches a peak in its local region. This is where the function value changes from increasing to decreasing. Imagine you are hiking, and you reach the top of a small hill. That hilltop would be a local maximum. To find these points, we rely on the first derivative:
1. **Derivative Zero:** When \(f'(x) = 0\), it suggests a potential local maximum.2. **Sign change:** If the first derivative changes from positive to negative at that point, it confirms a local maximum.
Keep in mind that local maxima are relative to their immediate surroundings. They are not necessarily the highest point of the entire function.

Local Minima

Local minima are the opposite of local maxima. These points represent the valley bottoms within a certain region of the function. Think of walking in a park and finding a low spot in the landscape—that would be a local minimum. Here's how we identify them:

- **Derivative Zero:** Just like with maxima, check where \(f'(x) = 0\) for potential minima.- **Sign change:** The first derivative should change from negative to positive at that point to confirm a local minimum.
Local minima, like local maxima, are only the lowest points within a close range of \(x\)-values, not necessarily for the entire function.

Undefined Derivative

An undefined derivative occurs where \(f'(x)\) does not have a value. This can happen in several scenarios, such as:

• **Cusps:** Sharp points where the slope of the tangent line abruptly changes.
• **Vertical Tangents:** Where the slope tends towards infinity.
• **Discontinuities:** Breaks or jumps in the function graph.

These points can be critical since they might indicate sharp features or breaks in the graph, which are key to understanding the function's behavior. Identifying where derivatives are undefined is necessary for a complete analysis of critical points.