1. 4.3 How Derivatives Affect the Shape of a Graph

2. Facts If f ’( x ) > 0 on an interval ( a,b ), then f (x) is increasing on ( a,b ). If f ’( x ) < 0 on an interval ( a,b ), then f (x) is decreasing on ( a,b ). Example Find the intervals where the function is increasing or decreasing.

3. The First Derivative Test Let c be a critical number of f (x).  If f ’(x) changes from + to – at c, then f has a local maximum at c.  If f ’(x) changes from – to + at c, then f has a local minimum at c.  No sign change at c means no local extremum (maximum or minimum)

4. How to find local max/min and interval of increasing/decreasing 1)Find all critical values by solving f ’( x ) = 0 and f ’( x ) = undefined 2)Put all critical values on the number line and pick some test values to determine the sign of the derivative for each interval. 3)Determine the interval of increasing/decreasing based on the sign of the first derivative.

5. Examples Find the intervals of increase/decrease and all local extrema.

6. Examples

7. Definition  A function f ( x ) is concave up on an interval ( a,b ) if the graph of f (x) lies above its tangent line at each point on ( a,b ).  A function f ( x ) is concave down on an interval ( a,b ) if the graph of f (x) lies below its tangent line at each point on ( a,b ).  A point where a graph changes its concavity is called an inflection point. o If the slopes of the tangent lines are increasing, then the function is concave up. o If the slopes of the tangent lines are decreasing, then the function is concave down. Facts

8. Test For Concavity  If f ’’( x ) > 0 on an interval ( a,b ), then f (x) is concave up on ( a,b ).  If f ’’( x ) < 0 on an interval ( a,b ), then f (x) is concave down on ( a,b ).  If f ’’( x ) changes signs at c, then f (x) has an inflection point at c. Example : Find the intervals where the function is concave up or concave down, and find all inflection points.

9. How to find inflection points and interval of concavity 1)Find all numbers c such that f ’’(c) = 0 or f ’’(c) is undefined. 2)Put all values found in step 1 on the number line and use test values to determine the sign of the second derivative for each interval. 3)Determine the interval of concavity based on the sign of the second derivative.

10. Examples Find the intervals of concavity and all inflection points.

11. Second Derivative test Let c be a critical number of a function f (x). If f ’’( c ) > 0, then f (x) has a local minimum at c. If f ’’( c ) < 0, then f (x) has a local maximum at c. If f ’’( c ) = 0 or dne, then this test fails  must use the first derivative test! Example : Find all local extrema

12. Examples a) Find the intervals where the function is increasing /decreasing, and find all local extrema. b) Find the intervals where the function is concave up or concave down, and find all inflection points.