Understanding the First Derivative Test: Identifying Function Behavior and Local Extrema

First Derivative Test (function behavior)

The First Derivative Test, also known as the function behavior test, is a method used to determine the behavior of a function using its first derivative

The First Derivative Test, also known as the function behavior test, is a method used to determine the behavior of a function using its first derivative. This test helps to identify where a function is increasing or decreasing, and where it has local extrema (maximum and minimum points).

To apply the First Derivative Test, follow these steps:

1. Find the first derivative of the given function.
2. Solve the equation f'(x) = 0 to identify the critical points (points where the derivative is zero or undefined) of the function.
3. Create a number line and place the critical points on it.
4. Choose a test point from each interval created by the critical points and check the sign of the derivative at that point.
– If the derivative is positive (+) at a test point, the function is increasing in that interval.
– If the derivative is negative (-) at a test point, the function is decreasing in that interval.

Here’s an example to illustrate the process:

Let’s consider the function f(x) = x^3 – 3x^2 + 2x.

Step 1: Find the first derivative:
f'(x) = 3x^2 – 6x + 2.

Step 2: Solve f'(x) = 0 to find the critical points:
3x^2 – 6x + 2 = 0.
Using the quadratic formula or factoring, we find that x = 1 and x = 2/3 are the critical points.

Step 3: Create a number line with the critical points:
(-∞, 2/3) | (2/3, 1) | (1, ∞)

Step 4: Choose test points and determine the sign of f'(x):
– Choose x = 0 for interval (-∞, 2/3):
f'(0) = 3(0)^2 – 6(0) + 2 = 2.
The derivative is positive, so the function is increasing in this interval.

– Choose x = 1/2 for interval (2/3, 1):
f'(1/2) = 3(1/2)^2 – 6(1/2) + 2 = -1/4.
The derivative is negative, so the function is decreasing in this interval.

– Choose x = 2 for interval (1, ∞):
f'(2) = 3(2)^2 – 6(2) + 2 = 2.
The derivative is positive, so the function is increasing in this interval.

Based on the First Derivative Test, we can conclude:
– The function is increasing on the interval (-∞, 2/3) and (1, ∞).
– The function is decreasing on the interval (2/3, 1).

Lastly, we can identify any local extrema (maximum or minimum points) by observing the changes in the function’s behavior at the critical points. In this case, we have two critical points: x = 1 and x = 2/3. Since the function is increasing before and after x = 1, it has a local minimum at x = 1. Similarly, since the function is decreasing before and after x = 2/3, it has a local maximum at x = 2/3.

Therefore, the function f(x) = x^3 – 3x^2 + 2x has a local maximum at x = 2/3 and a local minimum at x = 1.

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