This document discusses using the second derivative test to find relative extrema of a function. It explains that you first find the critical numbers where the first derivative is equal to zero or undefined. You then take the second derivative at those critical numbers. If the second derivative is positive, it is a relative minimum, and if negative, it is a relative maximum. The document provides an example of using this process to find the relative extrema of the function f(x) = -3x^5 + 5x^3, determining it has a relative minimum at x = -1 and a relative maximum at x = 1.

1. 3.4b Concavity and the
2nd Derivative Test
OBJECTIVES:
Determine intervals on which a function is concave upward or concave
downward.
Find any points of inflection of a function.
•Apply the Second Derivative Test to find relative extrema

2. Concept Map time: Work in groups of 3 to write down all the things you know
about derivatives so far. Write down your ideas of what they are, what you
use them for, why are they important?
Derivativ
e

3. GSP

5. Ex 4. p195 Using the Second Derivative Test to find relative
max or mins
Find the relative extrema of f(x) = -3x5 + 5x3
Find critical numbers first (f‘=0, f‘ undef. in domain of f)
f '(x) = -15x4 +15x2 = -15x2 (x2 -1)
So critical numbers are x = 0,-1,1
Using f “(x) = -60x3 + 30x, apply 2nd Derivative test.
Point on f(x) (-1, -2) (1, 2) (0, 0)
Sign of f”(x) f ” (-1) > 0 f “(1) < 0 f “(0) = 0
Conclusion Concave up so
relative min
Concave down so
relative max
Test fails

6. Looking on either side of x = 0, the first derivative is
positive, so at x = 0 is neither a max or min.
2
2
1
-1
-2
-3
rel max at x = 1
test failed
at x=0
Rel min at
x = -1
f(x) = -3×x5+5×x3

7. To summarize:
Concavity
•Find the second derivative f” and see what values of
x makes it zero or not continuous.
•Set up intervals with these values
•Test intervals – if f”>0 it is concave up. If f” < 0 it is
concave down in interval

8. 2nd Derivative test
•Look for critical numbers of FIRST derivative
•Evaluate SECOND derivative at critical numbers
•If f” > 0 then that critical number is relative min
•If f” < 0 then that critical number is relative max
•If f” = 0, then test fails and you’ll have to look at
1st Derivative test to determine max, min or
neither.

9. with a partner how to keep
these straight!

10. 3.4b p. 196/ 29-37 every
other odd, 45, 79-82