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.