The First Derivative Test is a powerful tool in calculus for finding local maxima and minima (collectively known as local extrema) of a function. It utilizes the sign changes of the first derivative to determine where a function is increasing or decreasing. If the first derivative changes from positive to negative at a critical point, the function has a local maximum there. Conversely, if it changes from negative to positive, the function has a local minimum.
Essentially, you're looking for points where the slope of the tangent line to the function's graph changes its sign. This happens at critical points where the derivative is either zero or undefined. By analyzing the intervals around these points, we can identify where the function attains its local extrema.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Critical Point | A. A point where the first derivative changes sign from positive to negative. |
| 2. Local Maximum | B. A point where the first derivative is zero or undefined. |
| 3. Local Minimum | C. A point where the first derivative changes sign from negative to positive. |
| 4. Increasing Interval | D. An interval where the first derivative is positive. |
| 5. Decreasing Interval | E. An interval where the first derivative is negative. |
The First Derivative Test helps us find ______ ______ of a function by analyzing the sign of the ______ ______ around ______ ______. If the derivative changes from positive to negative, we have a ______ ______. If it changes from negative to positive, we have a ______ ______.
Explain, in your own words, why finding critical points is an essential step in using the First Derivative Test to identify local extrema.
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