1. Analyzing f(x) and f’(x) /
Graphical Representations Among f(x) and f’(x)

2. 12. Finding Relative Extrema
The lowest or highest point on f(x)
Set f’(x) equal to 0
Solve for x
Create a sign chart for f’(x) to determine min or max
ex: f(x) = x^2
f’(x) = 2x - +
2x = 0
X = 0
Max -
Min

3. 13. Finding Absolute Extrema on a Closed Interval
The lowest or highest point on f(x) including endpoints
Find the relative extrema
Compare those points to your endpoints
Ex: find the absolute min for x^2 on the interval [-2,2]
f(x) = x^2
f’(x) = 2x X
2x = 0
X = 0 Y
f(x) has an absolute min at x = 0. -2 2 4

4. 14. Intervals of Increasing/Decreasing
f(x) is Increasing if:
f’(x) is positive
f(x) is Decreasing if:
f’(x) is negative

5. 15. Determining Concavity and Points of Inflection
f(x) is Concave Up if:
f’’(x) is positive
f(x) is Concave Down if :
f’’(x) is negative
f(x) has a point of inflection when:
f’’(x) = 0 and changes signs.
Ex: + + 2 - + 2

6. 16. Second Derivative Test
Substitute each critical point into the second derivative
If:
f’’>0 then there is a relative minimum
f’’<0 then there is a relative maximum
f’’=0 then the test fails

7. Justifications pt.1
Justification Chart for increasing/decreasing intervals of functions
Situation
Justification
f(x) is increasing on the interval from (a,b)
f(x) is increasing on the interval from (a,b) because f’(x) > 0
f(x) is decreasing on the interval from (a,b)
f(x) is decreasing on the interval from (a,b) because f’(x) < 0

8. Justifications pt.2
Justifications of relative maximums/minimums and points of inflections
Situation
Justification
Relative minimum
f(x) has a relative minimum at x=a because f’(x) change points from negative to positive when x=a
Relative maximum
f(x) has a relative maximum at x=a because f’(x) changes signs from positive to negative when x=a
Point of inflection
f(x) has a point of inflection at x=a because f’’(x) changes signs when x=a

9. Homework Analyzing F and the Derivative MC: 1, 6, 9-12 FRQ: 13-15
Graphical Representation Among F and the Derivative
MC: 2, 3, 6-9,12
FRQ: 14-16