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This section describes extrema of a function as points of interest (PoI) on a
graph.
Extrema of a function
Here is a video!
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In a similar way to the intercepts, we will often be interested in the extrema of
a function, that is the maximum and minimum values. Extrema come in
two variations in addition to being either a max or min; specifically there
are local (also referred to as relative) extrema, and global (aka absolute)
extrema.
Absolute Extrema
Absolute extrema are typically the easier to state simply; they are the (absolutely)
largest and smallest
(The terms ’largest’ and ’smallest’ are often ambiguous in
mathematics. Here, we mean the largest as in the most positive number (or the value
that is furthest right on a number line) possible, and the ’smallest’ means the number
furthest toward the negative side (or the furthest left on the number line) possible.)
values that a function attains in it’s entire domain. Note that the absolute minimum
can be positive, just as the absolute maximum can be negative. Moreover, one should
note that a function may not attain a maximum (or a minimum) value. For
example, on the domain of all real numbers has no maximum or minimum
value. The function has no maximum value but has an absolute minimum.
The function has a maximum but no minimum value. Even if the function
attains its maximum or minimum, even though the value of the max or min is
(certainly) unique, there can be more than one point that attain those
values.
For example, a function that constantly oscillates between and (such as ) attains it’s
maximum value (specifically the value ) and it’s minimum value (specifically
the value ) infinitely many times. It’s important to draw the distinction
between how many absolute maximum (or minumum) values there are (at
most one) and how many points attain an absolute maximum (or minimum)
which could be anything between 0 (no absolute extrema) and infinitely
many.
Local Extrema
The other type of extrema are the local (or relative) extrema. These are points that
represent extreme values in some small ’neighborhood’ of the function. Essentially
these are points that are at the bottom of a valley or at the top of a hill on the graph.
The formal definition is a bit intimidating, and locating these values can be
exceptionally challenging.
(In fact, this is one of the primary areas of
study in calculus 1 and comes down to figuring out the zeros of a related
function to the one you are investigating.) Nonetheless we will visit how to
find local extrema in some specific cases in the coming topics, and further
exploration of this aspect will be a major focus in calculus for those that
continue.
1 : What is a good explanation of what an extrema of a function is?
The
extrema are points where the function attains an important value, like a break-even
point of a profit function.The extrema of a function is the lowest or highest point
a function might attain over all real numbers.Extrema represent the maximum
and minimum values of a function within some defined interval of it’s domain.There aren’t always extrema of a function, so there is no good explanation.
Precalculus Algebra MAC1140