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Analysis of Functions: Increase, Decrease, and Concavity
Increasing and Decreasing Functions
decreasing
decreasingincreasing
Theorem
Theorem
How about an Example?Find the intervals on which is increasingand the intervals on which it is decreasing?
2
_ _ _ _ _ _+ + + + + +
How about another example?Find the intervals on which is concave upand the intervals on which it is concave down.
1 3/2+ + + + + - - - - - - + + + + +
Inflection points
Points of inflection occur at the points where:•the derivative equals zero•the derivative doesn’t exist•the function changes concavity
Warning !!!
Asymptotes frequently act as inflection points, so make sure you check concavity changes before making conclusion.
Let’s get all this togetherexample
Considering find the following :
0 2
+ + + + + +_ _ _
1
_ _ _+ + +
Find the intervals on which f(x) is increasing, decreasing, concave up, concave down and inflection point.