Use the definitions of increasing and decreasing to argue that is decreasing on and increasing on . Then use derivatives to argue the same thing.
The statement has been proven.
We have been given a function .
We have to use the definitions of increasing and decreasing to argue that this function is decreasing on and increasing on .
For a and b in the interval
Thus the function is decreasing in the interval
For a and b in the interval role="math" localid="1648442301380"
Now if then,
Thus the function is increasing in the interval
The derivative of the function is given by :
The function is always negative for x<0
The function is always positive for x>0
Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of and examine any relevant limits so that you can describe all key points and behaviors of f.
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