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

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?

Derivative

GSP

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) 4 2 2 2'( ) 15 15 15 ( 1)f x x x x x

0, 1,1x So critical numbers are

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

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 failedat x=0

Rel min at x = -1

f x = -3x5+5x3

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

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.

with a partner how to keep these straight!

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