Question

Use the first and second derivative tests to find the local min and max of the function. :D

Answer 1

\[1+3x^2-2x^3\]

Answer 2

\[f'(x)=-6x^2+6x\]

Answer 3

why both? one should be enough....

Answer 4

\[6x(-x^2+1)=0\]

Answer 5

im not sure how to do the second derivative test...

Answer 6

critical points are 0, 1

Answer 7

f(0)=min and f(1)=max

Answer 8

\[f''x=-12x+6\]

Answer 9

\(\large f'(x)=6x- 6x^2=6x(1-x)\)

Answer 10

x=1/2 for the second one...

Answer 11

oh ya thats right,

Answer 12

evaluate the cp on the second derivative... what do you get?

Answer 13

x=1/2

Answer 14

i mean critical numbers, not critical points....
evaluate the second derivative on those cr. #s... what do you get?

Answer 15

f''(0) = ???
f''(1) = ???

Answer 16

so i take 0 and 1 and plug them in?

Answer 17

Calle, I'm not sure how you figured out that the 0 was your min, I guess you used your smarty brain. :D
Another way to determine which one is a min, and which is a max, (the way that they want you to try), is by doing the second derivative test.

Plug your critical points into the second derivative,
If the result is positive, then that part of the graph is concave up, meaning you have a minimum point.
If the result is negative, then that part of the graph is concave down, meaning you have a maximum point.

Yah try that :D

Answer 18

can't i find the max and min from plugging in the first derivative critical points into the origional equation?

Answer 19

Ummmmmmmmmmm
I think that will USUALLY work, yes.
But they didn't want you to determine it that way, they want you to practice using the second derivative test! :)

Answer 20

so, the second derivative is -12x+6

Answer 21

Because what comes next in calc is, being able to graph a function like that without a calculator.
So you really want to know where it curves upward, and downward.
So you need inflection points.

Using the second derivative test, i think, is just to prepare you for the next step :D

Answer 22

So determine if f''(0) is + or -
and also check f''(1)
:D

Answer 23

how can i find the max and min from the second derivative?

Answer 24

DO what i just told you to do ! >:O lol

Answer 25

f''(0) = -12(0) + 6

Is that positive or negative? :D

Answer 26

f(0) is the max f(1) is the min

Answer 27

positive

Answer 28

the first and second derivative contradict each other? :(

Answer 29

f(0)=1+0-0=1
f(1)=1+3-2=2

I think that agrees with what you get when you plug it into the function :O
Did you punch some numbers wrong maybe? <:O
Maybe I did :D I'll take another look.