Roughly, ‘extreme values’ are the
$\,y\,$-values
of high/low points on a graph.
You also usually want to know the
$\,x\,$-value(s) where these high/low points occur.
The $\,y\,$-value of a highest point is called a MAXIMUM value.
The $\,y\,$-value of a lowest point is called a MINIMUM value.
![]() maximum value is $\,7\,$; it occurs at $\,x = 3\,$ |
![]() minimum value is $\,2\,$; it occurs at $\,x = 3\,$ |
The plural of maximum is maxima. The plural of minimum is minima. |
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![]() maximum value is $\,7\,$; it occurs at $\,x = 3\,$ |
![]() minimum value is $\,2\,$; it occurs at $\,x = 3\,$ |
A ‘kink’ is a sudden change in direction. |
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![]() maximum value is $\,7\,$; it occurs at $\,x = 3\,$ |
![]() minimum value is $\,2\,$; it occurs at $\,x = 3\,$ |
Imagine tracing a graph with a pencil. A ‘break’ is a place where you have to pick up the pencil. |
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![]() maximum value is $\,7\,$; it occurs at $\,x = 5\,$ |
![]() minimum value is $\,2\,$; it occurs at $\,x = 1\,$ |
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are at horizontal tangent lines, kinks/breaks, and endpoints. But, you don't HAVE to have an extreme value at such places! In all the examples below, there are greater and lesser values arbitrarily close by. |
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![]() endpoint; no extreme value here |
To the left of the endpoint, the function oscillates: it's $\,2\,$ for rational inputs, and $\,-2\,$ for irrational inputs. You have to work hard to get an example of an endpoint that's not a max or min! |
Suppose you have a
function, $\,f\,$.
A function is a ruleit takes an input, does something to it, and gives a corresponding output. The function $\,f\,$ can be viewed as a box:
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For extreme value questions, you have the following scenario: You're interested in a particular set of inputsit might be the entire domain; it might be some interval. You drop these inputs of interest into the function box, and get a corresponding pile of outputs. Take a look at this output pile:
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Sometimes you're interested in highest/lowest values overall (like at B, E, and F). These are called global (or absolute) max/min. Sometimes you're interested in ‘local’ highest/lowest values (like at A, C, and D). These are called local (or relative) max/min. These are highest/lowest, provided you don't look too far away! [Note: You'll learn in Calculus that a global max is also a local max, and a global min is also a local min.] Some people like the terminology global/local. Others like absolute/relative. You choose! Calculus provides language and tools to enable precise discussions of extreme values and related concepts. The informal language used here (e.g., ‘high/low points’, ‘breaks’, ‘kinks’) will all be made precise in Calculus. This section is only an informal introduction. |
The field of Operations Research is devoted to maximizing or minimizing real-world objectives.
Maximize profits.
Minimize costs.
Minimize the amount of material needed to construct a box with a specified volume.
Extreme values have extreme applicability!
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You may not have any extreme values. You might have a max, but not a min. You might have a min, but not a max. In the graph shown, the number $\,5\,$ is ‘trying to be’ the maximum value. But, there's no $\,5\,$ in the output pile, because the point isn't there! For example, you could pick up a $\,4.9\,$ from the output pilebut this wouldn't be greatest, because there's a $\,4.99\,$ there! And $\,4.99\,$ isn't greatest, because there's a $\,4.999\,$ there (and so on). There's an important theorem in Calculus, called the Max/Min Theorem (or the Extreme Value Theorem) that guarantees conditions under which you'll have both a max and a min. Calculus is so wonderful! |
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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