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Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa.

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Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa. -3 4

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Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa. -3 4 Relative Max.

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Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa. -3 4 Relative Max. Relative Min.

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Maxima and Minima of Functions

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INCREASING DECREASING

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Maxima and Minima of Functions

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Relative Minimum

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Maxima and Minima of Functions

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INCREASING DECREASING INCREASING

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Maxima and Minima of Functions Relative Maximum Relative Minimum

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Maxima and Minima of Functions

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The difference in this example is we are restricted to a specific interval. So the edges of the interval will act as critical points along with the ones we find using the first derivative. They will be relative max or min depending on their position.

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Maxima and Minima of Functions The difference in this example is we are restricted to a specific interval. So the edges of the interval will act as critical points along with the ones we find using the first derivative. They will be relative max or min depending on their position. Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum.

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum. Relative Maximum

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum. Relative Maximum This is neither because there is no change in increasing/decreasing. It is called an “inflection point” which we will discuss later…

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum. Relative Maximum Relative Minimum Change from decreasing to increasing…

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Maxima and Minima of Functions Moving left to right : If the edge has a decreasing arrow following, it is a relative maximum. If the edge has an increasing arrow following, it is a relative minimum. If the edge has a decreasing arrow in front of it, it is a relative minimum. If the edge has an increasing arrow in front of it, it is a relative maximum. Relative Maximum Relative Minimum Relative Maximum

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Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

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