Question

Calculus inflection and concavity?
find the points of inflection and discuss the
concavity of the graph of the function.

f(
x)= 2x^3 - 3x^2 -12x  5

Answer 1

Can you take the derivative of this?

Answer 2

yes

Answer 3

its 6x^2-6x-12

Answer 4

ok so do that and set it equal to zero the solutions are your points of inflection.

Answer 5

so 6x^2-6x-12=0

Answer 6

isn't it second derivative?

Answer 7

so solve that

Answer 8

this is what I wanted to confirm because I wasn't sure

Answer 9

no you can also just first take the first derivative to see where the points of inflection are and then take the second derivative to see if its concave up or down at these points.

Answer 10

so can you solve it?

Answer 11

6x^2-6x-12=0
6 (x^2-x-2)

Answer 12

is that right?

Answer 13

now after you are done with that take the second derivative of your equation which is 12x -6

Answer 14

yah i thts right

Answer 15

how do I find the points on that?

Answer 16

i know for sure x=0 right?

Answer 17

no y=0 because its y=6x^2-6x-12

Answer 18

and anyway the solutions for x are -1 and 2

Answer 19

now take the second derivative and plug in these two numbers in it

Answer 20

so 12(-1)-6
and 12(2)-6

so -18 and 18

Answer 21

see how one is positive and the other is negative?

Answer 22

the negative one is concave up and the positive one is concave down

Answer 23

so you can say that sometimes the function is concave up and sometimes its concave down

Answer 24

ok I see. One thing I don't understand though is why did you use y? I thought it was x or f(x)

Answer 25

because f(x) is the y axis

Answer 26

you could also think of it as just f(x)=0

Answer 27

so negative means concave up and positive means concave down

Answer 28

ok I understand. Can I also solve this by finding the second derivative solutions or do i have to find the first equation solutions then insert into second derivative always?>

Answer 29

@mihirb

Answer 30

To me, the first derivative is for critical points, the second derivative for inflection points and concavity only.

Answer 31

@Loser66 this is what I thought

Answer 32

so should I do follow the same steps except just use the second derivative?

Answer 33

i m with you

Answer 34

thank you for clearing that up

Answer 35

XD