Question

For y = f(x) = x^12 - 12x, and 0≤x≤2, find the value(s) of x for which....

1. f(x) has a local maximum... (write in ascending order!)
x=_______, x=_______

2. f(x) has a local minimum. If there are none, write NA.
x=_______

3. f(x) has a global maximum. If there are none, write NA.
x=_______

4. f(x) has a global minimum. If there are none, write NA.
x=________

Thank you!!

so would the first step be to get the derivative and set equal to zero? :/

Answer 1

take the derivative

Answer 2

okay:)

so \[f'(x)=12x ^{11} - 12\] ?

Answer 3

a local max or min occurs when f'(x) = 0

Answer 4

okay so

12(x^11 - 1) = 0 ?

is that how i would set it up?

Answer 5

yeahhhh

Answer 6

okay so

using x^11-1=0

you get x^11= 1

so \[x=\sqrt[11]{1}\] ? is that right? :/

Answer 7

haha yes

Answer 8

and also x=0 ?

so x= 0 and x= 1 ? :/

Answer 9

how do you get x=0?

Answer 10

oh so is it just x=1?

Answer 11

i wasn't sure, because my problem asks for two local maxes, but i only found x=1 :/ haha

Answer 12

x=1, is the point , but is is a max or min? is the second derivative (at x=1) positive or negative?

Answer 13

oh so second derivative is

\[f''(x)=132x ^{10}\] ?

Answer 14

yeah, now is f''(1) positive or negative?

Answer 15

if it's positive , it mean the graph is curving upwards, ie its a min
if negative , curving downwards, ie a max

Answer 16

positive right?

so it's a max?

Answer 17

um,

Answer 18

if it's positive , it mean the graph is curving upwards, ie its a min

Answer 19

ohh so it's a local min?

so our local min here is x=1 ?

Answer 20

yes

Answer 21

yay! :)

so what about the local maxes?

Answer 22

This graph can give you some ideas
plot x^12 - 12 x from x= -1.2 to x=1.4

Answer 23

http://www.wolframalpha.com/input/?i=plot+x%5E12+-+12+x+from+x%3D+-1.2+to+x%3D1.4

Answer 24

See what happens near x=1
http://www.wolframalpha.com/input/?i=plot+x%5E12+-+12+x+from+x%3D+-1.2+to+x%3D-.8

Answer 25

Sorry near x = -1

Answer 26

I agree there is only one local minimum at x=1

Answer 27

okay:)

so what about the local maxes? :/

how do we find those?

Answer 28

This function doe not have a local maximum.

Answer 29

You need f'(x)=0 and f''(x) < 0
The second derivative of your function is \(132 x^{10} \ge 0 \). So it cannot be negative

Answer 30

ohh so there are no local maximums?

my problem says that f(x) has a local maximum and asks to order them in ascending order... so there are two?

confused haha :P

Answer 31

There is no local maximum everywhere. I did not see that the interval is restricted to \( 0\le x \le 2\). Hold on, we have to consider the end points
0 and 2

Answer 32

okay:)

Answer 33

f(0)=0 is a local maximum and
f(2)=4072 is also a local maximum

Answer 34

ohh okay

so i would write

x=0, x=4072 ?

Answer 35

or is it x=0 and x=2 ?

Answer 36

No
x=0 and x=2

Answer 37

ahh okay:)

so 1 and 2 are done!

what about global max and min?

Answer 38

At x =0 f has a global minimum
at x=2 f has a global maximum

Answer 39

ohhh so global is still based off what we found from local information?

Answer 40

x=x0 is a global min means that for every x in [0,2]
\[
f(x) \ge f(x0)
\]

Answer 41

x=x0 is a global max means that for every x in [0,2]\[
f(x)\le f(x0)\]

Answer 42

ahh okay:) awesome!! so this problem is now complete? :O

Answer 43

Yes

Answer 44

awesome! thank you! :)

Answer 45

YW