Question

Find the absolute maximum of f(x) = 12x - x^3 on the closed interval [-4,1] show work please

Answer 1

find the first derivative, set it to zero. You should get some crtical points. Then test these point along with the endpoints.

Answer 2

The first derivaitve is: y'=12-3x^2
When we factor this we get: 3(4-x^2)
Further we see we have a differce of square leading to: 3(2-x)(2+x), lets set this to zero and we get:x=2, x=-2

Answer 3

Those would be critical numbers.

Answer 4

Now we test each critical number by plugging in a smaller number into derivaitve, then a sligthly larger number into the derivative. If we see a change from positive to negtive we have a maximum, if we see a change from negative to positive, we have a mininium. We have to do this will all the critical number including the endpoints.

Answer 5

how do i distinguish and absolute max from a relative max?

Answer 6

The highest max and lowest min (that pretty funny) is your absolute extrema, the other extrema are relative

Answer 7

ok thanks

Answer 8

so would it be total of 5 test points for 4 numbers?

Answer 9

-4 -2 1 2
------- --------- --------- ---------
-5 -3 -3 -1 0 1.5 1.8 4

Answer 10

The two numbers on the bottom of the critical numbers are those test values you plug into the deriviatve

Answer 11

right i just did that and on my number line from left to right got - - + + -

Answer 12

The answer in my sample exam is 16, how the heck do you get that?

Answer 13

are my signs right?

Answer 14

I think i got it plug 2 into original function for answer?

Answer 15

16, they might be asking for the y value (height)

Answer 16

yeah that sounds right

Answer 17

between + - would be absolute max?

Answer 18

There is a max at 2 and a min at -2

Answer 19

cool, would + + and/or - - be known as saddle point exclusively?

Answer 20

hold on let me test the critical points, just to make sure

Answer 21

okay, yeah, testing values for -2 we get - to +, testing values for 2 we get + to -

Answer 22

K thx... So what is + to + and - to - ?

Answer 23

Well inorder to have extrema we need to have a change from positive to negtaive through a zero.

Answer 24

or negative to positive