Given the graph of f of x,
determine the relative and absolute extrema.
Let's begin by reviewing the definitions
of absolute maximum and absolute minimum.
F of x has an absolute or global maximum
of f of c at x equals c
if f of c is greater than or equal to f of x
for all x in the domain,
which means we can identify the absolute maximum
if it exists, by identifying the highest point
of all of the points on the graph.
Analyzing the graph,
notice how the highest point is this point here,
where the ordered pair is negative 4 comma 5,
which indicates that the y-value, or function value, of 5
is the absolute maximum,
which occurs at x equals negative 4.
We list 5 as the absolute maximum.
F of x has an absolute or global minimum of f of c
at x equals c if f of c is less than equal to f of x
for all x in the domain,
which means we identify the absolute minimum,
if it exists, by identifying the lowest point
of all of the points on the graph.
Notice how the lowest point on the graph
is the left end point, this closed point here,
where the ordered pair is negative 7 comma negative 4.
The y-value, or function value, of negative 4
is the absolute minimum,
which occurs at x equals negative 7.
And now let's review the definitions of relative maximum
and relative minimum.
F of x has a relative or local maximum
of f of c at x equals c
if f of c is greater than or equal to f of x
for all x in some open interval containing x equals c.
So we identify the relative or local maximum
by identifying the high points on the graph.
However, notice x equals c
must be in some open interval in the domain.
This means you must be able to approach x equals c
from the left and right,
and therefore, if we have closed end points,
these cannot be considered for relative extrema,
or in this case, the relative maximums.
Similarly, f of x has a relative or local minimum
of f of c at x equals c
if f of c is less than equal to f of x
for all x in some open interval containing x equals c.
We identify the relative minimums
by identifying the low points on the graph
of the given function.
However, once again, we have closed endpoints.
We can't include the endpoints
as possible relative minimums.
Again, this is because we cannot approach the endpoints
from the left and right, which is what's necessary
in order for x equals c to be in some open interval.
So going back to the graph,
we must exclude the left end point
as a possible relative minimum.
We must also exclude the right end point
as a possible relative maximum,
again, because we're not able to approach the endpoints
from the left and right.
And now to identify the high and low points
on the graph.
I'll notice the point negative 4 comma 5 is a high point,
and there is an open interval in the domain
containing x equals negative 4,
meaning we can approach x equals negative 4
from the left and right.
So because we have a high point,
the y-value, or function value, of 5 is a relative maximum,
which occurs at x equals negative 4.
So notice how it is possible for a point
to represent an absolute maximum
as well as a relative maximum.
Also notice how the function changes
from increasing to decreasing at this high point,
which results in a relative max.
Next, it appears as if we have a low point
at the point negative 2 comma 1,
but notice how this is an open point,
which indicates the function is undefined at this point,
and therefore, this is not a low point on the graph.
Next, notice how we have a high point here,
where we'll say the ordered pair is negative 0.8 comma 3.
Notice how there is an open interval
containing x equals negative 0.8.
And this point is a high point.
And therefore the y-value, or function value, of 3
is a relative maximum,
which occurs at x equals negative 0.8.
We list 3 as a relative or local maximum.
And once again, notice how the function does change
from increasing to decreasing
at this point that represents a relative maximum.
Next, we have a low point here,
where the ordered pair is, let's say, 2.3 comma negative 3.
Notice there is an open interval containing x equals 2.3.
And this point is a low point.
And therefore the y-value, or function value,
of negative 3 is a relative minimum,
which occurs at x equals 2.3.
We list negative 3 as a relative minimum.
Also notice how the function changes
from decreasing to increasing
at this point that represents a relative minimum.
Next we have a high point here,
where the order pair is, let's say, 5.6 comma 3.
There is an open interval containing x equals 5.6.
This point is a high point, and therefore,
the y-value, or function value, of 3 is a relative maximum,
which occurs at x equals 5.6.
Notice how we already listed 3 as a relative maximum,
and therefore we don't list it again.
The relative maximum of 3 does occur at two locations,
at x equals negative 0.8 and x equals 5.6.
And once again, at this point,
notice how the function changes
from increasing to decreasing.
Now we have one more point to consider.
Notice how we have a closed point here,
which is a low point,
and the ordered pair for this point is 7 comma 0.5.
Even though the function does have jump discontinuity
at x equals 7,
there's an open interval containing x equals 7,
and this point is a low point,
and therefore the y-value, or function value, of 0.5
is a relative minimum.
Notice at this location,
the function does change from decreasing to increasing.
I hope you found this helpful.