Question

Describe the concavity of the graph of f on the interval [0, π].

f(x)=3x^2-3sin(2x)

Answer 1

You need to find \[f\prime \prime(x)\]
and look at this function over the range [0,pi]

when the function is positive/negative it means positive/negative concavity

Answer 2

I know the derivative of it is 6x-6cos2x but when I equal it to zero

\[x-\cos2x=0\]

Answer 3

Oh so I dont need to pay attention for the first derivative

Answer 4

No not for concavity. First derivative test is to find the extreme points of f(x)

Answer 5

Oh ok then the second derivative is

\[6+12\sin(x)\]

So then I equal that to zero correct?

Answer 6

I ment the second derivative is
\[6+12\sin(2x)\]

Answer 7

My bad

Answer 8

Yes set to zero and look for positive and negative values

Answer 9

So far I got
\[\sin(2x)=\frac{ -1 }{ 2 }\]

Answer 10

No sorry not set to zero... you plug in the domain [0, pi] and see when it is positive or negative

Answer 11

ok then its positive from 0 to pi/2 then negetive at 2pi/3 to 5pi/6 then positive at pi

Answer 12

But both pi and 0 are positive

Answer 13

Im confused because my awnser choices are

a)concave up on (0, 7π⁄12 ); concave down on ( 7π⁄12, π).
b)concave up on ( 7π⁄12, 11π⁄12 ); concave down on (0, 7π⁄12 ) and on ( 11π⁄12, π).
c)concave down on (0, π).
d)concave up on (0, 7π⁄12 ) and on ( 11π⁄12, π); concave down on ( 7π⁄12, 11π⁄12 ).
e)concave up on (0, π⁄12 ); concave down on ( π⁄12, π).

Answer 14

I graphed the function and found
positive (concave up) from (0, 7pi/12)
negative (concave down) from (7pi/12, 11pi/12)
positive (concave up) from (11pi/12, pi)

Answer 15

I think you were missing some number when you performed the last step

Answer 16

http://www.wolframalpha.com/input/?i=6%2B12sin%282x%29

You can see a graph here of the function

Answer 17

I guess if I needed to do it without a calculator I needed to solve the second derivative

Answer 18

and set it as equal to zero correct?

Answer 19

No you dont need to set the second derivative to zero because that does not tell you any information. Only if the second derivative is positive or negative does it tell you the concavity.

You set the first derivative equal to zero to find the extreme points of the original function.

Answer 20

Another type of question with this is to find the extreme points of f(x) with the first derivative test

and then use the second derivative at those points to see if they are a max( negative concavity) or a minimum (positive concavity)