Analyzing Function Behavior: The First Derivative Test for Local Extrema and Monotonicity

First Derivative Test (function behavior)

The first derivative test is a method used to analyze the behavior of a function based on its first derivative

The first derivative test is a method used to analyze the behavior of a function based on its first derivative. It helps determine the local extrema (maximum and minimum points) of a function and the intervals where the function is increasing or decreasing.

To apply the first derivative test, follow these steps:

1. Find the first derivative of the given function.
2. Set the first derivative equal to zero and solve for x. The values of x obtained here are the critical points of the function.
3. Create a number line and mark the critical points on it. These critical points divide the number line into intervals.
4. Choose a test point from each interval and plug it into the first derivative. Evaluate whether the function is positive or negative in that interval.
5. Summarize the results in a table or graphical format to determine the behavior of the function.

Now let’s go through an example to better understand how the first derivative test is applied.

Example:
Consider the function f(x) = x^3 – 3x^2 – 9x + 5.

Step 1: Find the first derivative of f(x).
f'(x) = 3x^2 – 6x – 9

Step 2: Set f'(x) = 0 and solve for x.
3x^2 – 6x – 9 = 0
Dividing both sides by 3:
x^2 – 2x – 3 = 0
Factoring:
(x – 3)(x + 1) = 0
Therefore, x = 3 and x = -1 are the critical points of the function.

Step 3: Create a number line and mark the critical points on it.
-1 3

Step 4: Choose test points from each interval and evaluate the sign of f'(x) in those intervals.
For interval (-∞, -1):
Choose x = -2:
f'(-2) = 3(-2)^2 – 6(-2) – 9 = 12 + 12 – 9 = 15 > 0
Therefore, f'(x) is positive in this interval, indicating that the original function f(x) is increasing.

For interval (-1, 3):
Choose x = 0:
f'(0) = 3(0)^2 – 6(0) – 9 = -9 < 0 Therefore, f'(x) is negative in this interval, indicating that the original function f(x) is decreasing. For interval (3, ∞): Choose x = 4: f'(4) = 3(4)^2 - 6(4) - 9 = 48 - 24 - 9 = 15 > 0
Therefore, f'(x) is positive in this interval, indicating that the original function f(x) is increasing again.

Step 5: Summarize the results in a table or graphical format.
Interval Sign of f'(x) Behavior of f(x)
(-∞, -1) + Increasing
(-1, 3) – Decreasing
(3, ∞) + Increasing

From this analysis, we can conclude that the function f(x) has a local minimum at x = -1 and a local maximum at x = 3.

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