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Recognizes the absolute extrema & relative extrema for the following function.
f ( x ) = x2 on [-1, 2]
Solution: As this function is simple enough to graph let's do that. Though, we only want the graph on the interval [-1,2]. Here is the graph,
Note as well that we utilized dots at the end of the graph to remind us that the graph ends at these points.
Now we can identify the extrema from the graph. It looks like we've got a relative & absolute minimum of zero at x = 0 and an absolute maximum of four at x = 2 . Note as well that x = -1 is not a relative maximum as it is at the ending point of the interval.
This function doesn't contain any relative maximums.
As we saw in the previous example functions do not have to have relative extrema. It is entirely possible for a function to not have a relative maximum and/or a relative minimum.
Quotient Rule : If the two functions f(x) & g(x) are differentiable (that means the derivative exist) then the quotient is differentiable and,
Extrema : Note as well that while we say an "open interval around x = c " we mean that we can discover some interval ( a, b ) , not involving the endpoints, such that a Also,
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