Mastering the First Derivative Test: Analyzing Function Behavior and Locating Extrema

First Derivative Test (function behavior)

The First Derivative Test is a technique used to analyze the behavior of a function using its first derivative

The First Derivative Test is a technique used to analyze the behavior of a function using its first derivative. It helps determine the increasing and decreasing intervals of a function, as well as the location of its local extrema (maximum and minimum points).

To apply the First Derivative Test, follow these steps:

1. Find the first derivative of the function.
– If the function is given in the form f(x), differentiate it with respect to x. Let’s call this derivative f'(x).

2. Find the critical points (where f'(x) = 0 or f'(x) is undefined).
– Set f'(x) = 0 and solve for x.
– Also, check for any values of x where f'(x) is undefined (usually caused by division by zero or square roots of negative numbers).

3. Determine the intervals of increasing and decreasing.
– Use a number line and mark the critical points and any points where f'(x) is undefined.
– Choose a test value from each interval formed on the number line and evaluate f'(x) at that point.
– If f'(x) is positive, the function is increasing on that interval.
– If f'(x) is negative, the function is decreasing on that interval.

4. Analyze the local extrema.
– If a critical point x = c falls within an interval where the function is increasing, it represents a local minimum.
– If a critical point x = c falls within an interval where the function is decreasing, it represents a local maximum.

To summarize:
– f'(x) > 0 indicates an increasing function.
– f'(x) < 0 indicates a decreasing function. - A critical point where f'(x) changes from positive to negative represents a local maximum. - A critical point where f'(x) changes from negative to positive represents a local minimum. It's important to note that the First Derivative Test doesn't tell us anything about the behavior of the function outside of the critical points, so there might be additional maximums or minimums present. To find the overall maximum or minimum of a function, you would need to consider the behavior at the function's endpoints and other critical points (such as when the derivative is undefined). Remember to re-check your calculations and consider any conditions or constraints given in the problem you are working on.

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