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

- anonymous

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- anonymous

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

- anonymous

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

- anonymous

DO WE FIND ALSO 2ND DERIVATIVE TOO?

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## More answers

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

i dont believe 2nd derivative is necessary for determining extremas

- anonymous

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

- anonymous

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

- anonymous

ok show it.

- anonymous

why do you multiply it?

- anonymous

and the middle term is -122x

- anonymous

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

- anonymous

yes. right..

- anonymous

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

- anonymous

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

- anonymous

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.

- anonymous

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.

- anonymous

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

- anonymous

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?

- anonymous

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

- anonymous

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

- anonymous

since 122/2 = 61..

- anonymous

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

- anonymous

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

- anonymous

There you go.

- anonymous

which one is max and min?

- anonymous

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

- anonymous

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.

- anonymous

O_o

- anonymous

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

- anonymous

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

- anonymous

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.

- anonymous

|dw:1363299185209:dw|

- anonymous

Hope that makes sense.

- anonymous

f(0)= 3 and f(4) -589

- anonymous

okay those are the extrema points.. but how about the work for x= 8 and x= 2/15?

- anonymous

@calmat01

- anonymous

its wrong....

- anonymous

i see how you got the 16 and -232... but those aren;t the min or max of the interval.

- anonymous

i put 16 as min ad -232 as max. its wrong.

- anonymous

No, I was telling you that the max occurs at x=2/15, not that the values I used were the max or min. You still have to determine the max by substituting your value for x=2/15.

- anonymous

Because the graph at x=2/15 changes direction...it goes from increasing to dercreasing thus identifying a max.

- anonymous

okay so far i have f'(0)=16 f'(4)=-232 and the x=8 and the other x= 2/15

- dan815

take first dervivative and find crit points and plug them into 2nd derivative to see if those crit ppts are maxima(negative for 2nd derv) or minima(positive)

- anonymous

i would appreciate if i can see some work please.

- dan815

do u understand what i said

- anonymous

yeah but i'm more of a visual person..

- anonymous

i understand what you're saying though.

- dan815

http://www.wolframalpha.com/input/?i=16-122+x%2B15+x%5E2&lk=1&a=ClashPrefs_*Math-

- dan815

that is the graph of ur derivative

- dan815

to be honest if ur visual u should realize its a 3rd degree func to 2nd degree so its like this|dw:1363301376615:dw|

- dan815

so say you find f''(x) = 0 then you will know anything grreater than that x value will give u minimas and anything less than 0 for 2nd derv will give you maximas because they are inflection points

- dan815

these are inflection points say for your base function x^3 ...+...+...=y

- dan815

|dw:1363301676110:dw|

- dan815

so whats its basically asking you is to find the critical pts and tell you if they are concave up or down so for the example above u know the critical pts are these|dw:1363301848503:dw|

- dan815

where your slope is 0 right so those are crit pts but do those crit points lie where the graph is concave up or down

- dan815

for the first crit pt you can see its on the part of the graph where it is concave down and the 2nd one is on the part of the graph where its concave up

- dan815

make sense now?

- dan815

but lets suppose this is your interval |dw:1363302163817:dw|

- dan815

so for the first critical point you know thats a maxima between the interval where your interval is x between those 2 vertical lines, but your minimum in that interval wont be the 2nd critival point because that is outside of the interval rather you will have to check the ends of the interval to see which one is of less value and that is your minumum

- anonymous

oh okay thank you=]

- anonymous

i see what method you used.

- anonymous

so is the minim here: -589 and the max -1213??

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