Question

Find where the function is concave up: f(x) = 3x4 - 2x3 – 3x

Answer 1

So far I did find f '(x)which equals 12x^3-6x^2-3

Answer 2

f'' is a good test for cave up....which is a MIN

find your "zeros" for the f' and that will tell you what "x" values you need to use for the next part..

Answer 3

3(4x^3 -2x^2 -1) narrows it down.

4x^3 -2x^2 -1 = 0

x^2(4x-2) = 1 might help

Answer 4

lets go ahead and find the f'' might be easier

Answer 5

f'' = 36x^2 - 12x now we can partition this out to determine what values we get for the number line

Answer 6

12x(3x -1).... we have zero values at 0 and 1/3

these are out inflections if were lucky

<........0.........1/3..........>
- + +
- - +
----------------------
+ - +

getting warmer

Answer 7

0 and 1/3 are our inflections were the concavity changes

(-inf,0) = cave up
(0,1/3) = cave down
(1/3, inf) = cave up

so this thing looks like a "W"