Calculus 2nd Derivative test

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