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Max and Min Values
Max/Min Values
~ The relative (local) maximum value of a function is the largest (y) that the function takes at a given point (x) excluding the endpoints.
~ The relative (local) minimum value of a function is the smallest (y) that the function takes at a given point (x) excluding the endpoints.
~ A graph may contain several relative max/min values, but contain only one absolute max/min point.
~ An absolute max/min point will be the highest/lowest point of all the relative max/min points on the graph.
~ These values are also identified as the relative (local) extrema and absolute extrema.
~ Because we will be graphing a function, we will not find the extrema using math.
We will determine the relative max/min points by looking at the highest and lowest points of the function's graph.
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Important!!! Make a note of this in your notebook.
Make sure you are aware of the difference between these statements.
~ When asked to find the VALUES of f(x), we are looking at f(x) -> the y-value.
~ When asked to find the coordinates of the max/min, we are looking for (x, y).
~ When asked to find WHERE the max/min occurs, we are looking at the x-value.
Example 1:
Determine the relative max and relative min values of f(x) = 2x3 -3x2 - 3.
Looking at the graph of f(x), it appears that the max's coordinates are (0, -3) and its min's coordinates are (1, -4).
***The relative max value is f(0) = 3, and the relative min value is
f(1) = -4.
Example 2:
Determine the relative max and min values of
f(x) = -2x4- 4x3 + x - 2.
Looking at the graph of f(x) (right), there are two relative max values and one relative min value.
In this case, the coordinates of the relative max are (-1.44, -0.096) and (0.266, -1.819), and its relative max values are f(-1.44) = -0.096 and f(.266) = -1.819.
***The coordinates of the relative min are (-0.326, -2.21) and its relative min values are f(-0.326) = -2.21.
***There is an absolute max value, and that occurs when x = -1.44, so the absolute max is f(-1.44) = -0.096.
This is a link to an activity that will check your understanding of the above concepts.
Click on each blue and red dot, and select the description that best fits that location.
After tagging all the dots, click the blue check circle at the bottom right of the screen to check your answers.
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