Question

H-E-L-P Determine the extrema of below on the given interval
f(x)=5x^3-61x^2+16x+3

(a) on [0,4]
The minimum is and the maximum is .
(b) on [-9,9]
The minimum is and the maximum is

Answer 1

first derivative and set to zero to find the local max and min
other points of interest would be the end points

Answer 2

THE DERIVATIVE IS 15X^2-122X+16

Answer 3

DO WE FIND ALSO 2ND DERIVATIVE TOO?

Answer 4

lol..here we are again...it's back to factoring.

Answer 5

You need to find factors of 240, which is the product of 15 and 16 which add to the coefficient of the middle term.

Answer 6

If you split the -122x into -120x and -2x, you will get the numbers I was referring to. Then you can factor by grouping.

Answer 7

yup here we are... where the 240 come from...

Answer 8

i dont believe 2nd derivative is necessary for determining extremas

Answer 9

http://www.sophia.org/finding-extrema-on-an-interval--2/finding-extrema-on-an-interval--8-tutorial

Answer 10

I explained already...240 comes from multiplying the 15(coefficient of squared term) and 16 (the constant).

Answer 11

ok show it.

Answer 12

why do you multiply it?

Answer 13

and the middle term is -122x

Answer 14

Because it is not a trinomial with leading coefficient of 1.

Answer 15

yes. right..

Answer 16

That is why I said you want numbers which add to the coeffcient of the middle term.

Answer 17

The two numbers which multiply to give you 240 that add to -122 are -120 and -2.

Answer 18

So, basic algebra tells us that if you want to factor an equation of the form \[ax ^{2}+bx+c=0 \] we must multiply a*c and find factors of a*c which add to the coeffcient of the middle term, in this case -122.

Answer 19

And those factors, as I already mentioned, are -120 and -2. So, just attach the x to each of them, group the first two and last two terms in separate parenthesis and factor by grouping.

Answer 20

\[15x ^{2}-122x+16=15x ^{2}-120x-2x+16=0\] Grouping them as I have instructed, you get \[(15x ^{2}-120x) + (-2x+16)=0\]

Answer 21

Now just factor a 15x from the first parenthesis, and a -2 from the second parenthesis, and what do you notices when you do that?

Answer 22

oh i got it... you just separated them ..

Answer 23

can you separate them by anything? ex: -61x ?

Answer 24

since 122/2 = 61..

Answer 25

nvm i understand.. you want something you can easily factor with 15 and 2

Answer 26

so i get.. x= 8 and 2/15

Answer 27

There you go.

Answer 28

which one is max and min?

Answer 29

wait the intervals are [0,4] what do we do with it?

Answer 30

Check to see what happens to the derivative to the left and right of the point x=2/15 since x=8 is not in the interval.

Answer 31

O_o

Answer 32

true.. but what do you mean derivative to the left and right?

Answer 33

plug x=0 and x=4 into the original equation in order to find the extrema points

Answer 34

You can tell whether you have a max or min by what happens to the derivative to the left and right of the critical point. If f'(x) > o, your function is increasing, if f'(x)<0, your function is decreasing.