## hiramoby Group Title Find all relative extrema of the function F(X)=16/(X^2+1). Use the Second-Derivative Test when applicable. Answer a. The relative maximum is .1,0 b. The relative minimum is .0,1 c. The relative maximum is .0,16 d. The relative minimum is .0,16 e. The relative maximum is .16,0 2 years ago 2 years ago

1. Australopithecus

Take the derivative of this function first

2. hiramoby

here we go

3. Australopithecus

First set f'(x) = 0 then solve for x, those will give you your critical points

4. hiramoby

I got 0

5. hiramoby

-32

6. Australopithecus

also make sure to check the domain of f'(x) and the domain of f(x) if something is not in the domain of f'(x) but is in the domain of f(x) then it is a critical point if a point is not in the domain of f(x) but is in the domain of f'(x) then it is not a critical point

7. hiramoby

so it is going to be e then

8. Australopithecus

ok make a table |dw:1351050419295:dw|

9. Australopithecus

check in between these numbers and right if they are positive or negative

10. hiramoby

11. Australopithecus

you need to check to see if they are maximums or minimums

12. Australopithecus

and I'm showing you how

13. Australopithecus

Oh I made a mistake the table should be |dw:1351050692097:dw| so the general rule is 1. If f'(x) > 0 on the Interval than f(x) is increasing on the interval 2. If f'(x) < 0 on the Interval than f(x) is decreasing on the interval

14. Australopithecus

So to show you an example of this -32x/(x^4 + 2x^2 + 1) sub in a number in the interval (-infinity, 0) so I'm going with -1 -32(-1)/((-1)^4 + 2(-1)^2 + 1) = a positive number so we add in the table |dw:1351050909010:dw|

15. Australopithecus

then we do it again for the next interval (0, -32) so I will pick 1 f'(1) = -32(1)/((1)^4 + 2(1)^2 + 1) = a negative number |dw:1351051038432:dw| so we see from this table that 0 is a maximum

16. Australopithecus

I mean minimum

17. Australopithecus

we can insert 0 into f(x) to see where this minimum is f(0) = 16/(X^2+1) f(0) = 16/1 f(0) = 16 so we have a minimum at (0,16)

18. Australopithecus

I hope this was helpful :S

19. Australopithecus

oh crap made a mistake lol |dw:1351051431080:dw| it should be this you are right it is e maximum at (0,16)