This document discusses concepts related to using the first and second derivatives to analyze functions and graph their behavior, including: - Concave up and down shapes defined by increasing/decreasing rate of change - Inflection points occurring where concavity changes - The second derivative indicating the rate of change of the rate of change (concavity) of the original function - Using the second derivative test to determine if critical points are relative maxima or minima based on the concavity. It provides examples of applying these concepts to specific functions through determining critical points, inflection points, intervals of increasing/decreasing behavior, and concavity. It also demonstrates graphing functions based on this analysis

1. 3.3 Graphing Using the First
and Second Derivatives
• What does concave up or concave down look
like?
• What is an inflection point?
• What does the second derivative tell us about the
original function f?
• What’s “The Second Derivative Test”?

2. A. What does concave up or
concave down look like?
• If I gave you a riddle that sounded like,
“The rate of increase is increasing,” or
“The slope of the tangent is increasing,”
could you draw a picture of that?
• That is CONCAVE UP!

3. • If I gave you a riddle that sounded like,
“The rate of increase is decreasing,” or
“The slope of the tangent is decreasing,”
could you draw a picture of that?
• That is CONCAVE DOWN!

4. B. What is an inflection point?
• An inflection point is where the concavity
changes (from up to down, or down to up).

5. C. What does the second derivative
tell us about the original function f?
• Recall that the first derivative gave us the
increase/decrease for f, essentially the
rate of change for f.
• Well, the second derivative will give us the
rate of change for the rate of change for f.
In other words, the rate of increase of the
rate of increase of f.
• Hey! That is concavity!

6. • So, remember when you found values for
which the first derivative equals zero or is
undefined, and you found critical points?
• When you take the second derivative and
find values for which it is zero or
undefined, you will be finding inflection
points!
• (That sounds like the beginning of a “sign
diagram”? Right, you are!)

7. A second derivative sign diagram….
• Find the second derivative.
• Set its numerator and denominator equal to zero,
factor, and solve to find inflection points.
• Chop up the number line into pieces at the
inflection points, and choose a test value for each
interval.
• Plug in each test value into the second derivative,
only caring if it will be positive, negative, or zero.
• Positive second derivative means f is concave up.
• Negative second derivative means f is concave
down.
• A second derivative that is ZERO means that f is
STRAIGHT (no concavity).

8. I might be asked to graph
something…
• What would it look like if f’ > 0 and f’’ > 0?
• What about f’ > 0 and f” < 0?
• What about f’ < 0 and f” < 0?
• What about f’ < 0 and f” > 0?

9. Graph f ( x ) = x 3 − 9 x 2 + 24 x showing all relative extreme (max/min) points
and all inflection points. (Let' s do a sign diagram for f' first, and then one for f' '.)

10. Besides the graph, I could have
asked you for:
• Critical numbers: 2, 4
• Intervals of increase of f: (-inf, 2),(4, inf)
• Intervals of decrease of f: (2, 4)
• Relative min: (4, 16)
• Relative max: (2, 20)
• Inflection pts: (3, 18)
• Intervals where f is concave up: (3, inf)
• Intervals where f is concave down: (-inf, 3)

11. Graph f ( x ) = 18 x1/ 3 showing all relative extrema and inflection points.

12. Besides the graph, I could have
asked you for:
• Critical numbers: 0
• Intervals of increase of f: (-inf, inf)
• Intervals of decrease of f: none
• Relative min: none
• Relative max: none
• Inflection pts: (0, 0)
• Intervals where f is concave up: (-inf, 0)
• Intervals where f is concave down: (0, inf)

13. You try : f ( x ) = x + 3 x − 9 x + 5 Graph, show all
3 2
relative extrema and inflection points.

14. You try : f ( x ) = 9 x Graph, show all
4/3
relative extrema and inflection points.

15. D. What’s “the second derivative
test”?
• You may think it silly that we use the second
derivative to do all this stuff and THEN we have
some rule called “THE second derivative test.”
• (Consider you have a nice, smooth, normal, non-
pointy, non-asymptote-ey curve,)
• Imagine a relative maximum. What concavity
does it have there? CONCAVE DOWN!
• What about a relative minimum? CONCAVE UP!

16. THE second derivative test:
If x = c is a critical number of f at which f ′′ is defined,
(saying f ′′ is restricting this rule to only those nice,
non - cuspy, non - asymptote - ey places), then
f ′′( c ) > 0 means that f has a relative minimum at x = c.
f ′′( c ) < 0 means that f has a relative maximum at x = c.

17. Using the second derivative test:
• Take the first derivative and find critical
numbers.
• Plug the critical numbers into the second
derivative.
• If the second derivative is negative, there is a
rel. maximum at that x-value. (think concave
down)
• If the second derivative is positive, there is a rel
minimum at that x-value. (think concave up)
• If you get zero, then we don’t know what it is! It
could be a max or a min or an inflection point!
• Plug the critical numbers back into the original f
function to find their partnering y-values.

18. Use the second - derivative test to find all relative
extrema points of f ( x ) = x 3 − 9 x 2 + 24 x
(Polynomials will always have defined second derivatives.)

19. You try : Use the second - derivative test to find all relative
extrema points of f ( x ) = x 3 − 3 x 2 + 3 x + 4.