Maxima and Minima of Functions

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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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Presentation transcript:

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

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

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

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

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions DECREASING INCREASING

Maxima and Minima of Functions

Maxima and Minima of Functions Relative Minimum

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions

Maxima and Minima of Functions INCREASING DECREASING INCREASING

Maxima and Minima of Functions Relative Maximum Relative Minimum

Maxima and Minima of Functions

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.

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.

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.

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.

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.

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.

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.

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.

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.

Maxima and Minima of Functions Relative Maximum 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.

Maxima and Minima of Functions Relative Maximum This is neither because there is no change in increasing/decreasing. It is called an “inflection point” which we will discuss later… 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.

Maxima and Minima of Functions Relative Maximum Relative Minimum Change from decreasing to increasing… 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.

Maxima and Minima of Functions Relative Maximum Relative Minimum Relative Maximum 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.