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Extrema
The maximum and minimum value of an interval.
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Extreme Value Theorm
If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.
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Relative extrema
The highest or lowest point on a "hill".
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Critical numbers
A point at which f'(x) equals 0 or is undefined.
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How to find extrema on a closed interval:
- 1. Find critical numbers.
- 2. Evaluate f at each critical number in the interval.
- 3. Evaluate each endpoint of the interval (if it's closed)
- 4. The least is the minimum. The greatest is the maximum.
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Rolle's Theorem
If f is continuous on the closed interval and differentiable on the open interval and if the endpoints have the same y value, there is at least one number c in the interval such that f'(c) = 0.
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Mean Value Theorm
If f is continuous on the closed interval and differentiable on the open interval, then there is a tangent line whose slope is the same as the slope between the two endpoints (or the secant line).
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Increasing and Decreasing Functions
- 1. If f'(x) > 0 for all x in an interval, then f is increasing on that interval.
- 2. If f'(x) < 0 for all x in an interval, then f is decreasing on that interval.
- 3. If f'(x) = 0 for all x in an interval, then f is constant on that interval.
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First Derivative Test
If f'(x) changes from neg/pos to pos/neg at c, the there is a relative max or min at c.
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