## anonymous 4 years ago What are the coordinates of the inflection point on the graph of y=(x+1)arctanx a. (-1,0) b. (0,0) c. (0,1) d. (1, pi/4) e. (1, pi/2) work plzz :)

1. amistre64

2nd derivative is?

2. anonymous

i couldn't get that far :(

3. anonymous

i hav the first deriv

4. amistre64

we can go from there then, whats the 1st D ?

5. anonymous

f'(x)=1(1/1+x^2)

6. amistre64

hmmm, lets see if I can verify that: $y=(x+1)tan^{-1}(x)$ $y'=(x+1)'tan^{-1}(x)+(x+1)tan'^{-1}(x)$ $y'=tan^{-1}(x)+\frac{x+1}{1+x^2}$ $y'=tan^{-1}(x)+(x+1)(1+x^2)^{-1}$ you agree?

7. anonymous

ohhhhhhhhhhhh quotient rule!!

8. amistre64

$y'=tan^{-1}(x)+(x+1)(1+x^2)^{-1}$ $y''=tan'^{-1}(x)+(x+1)'(1+x^2)^{-1}+(x+1)(1+x^2)'^{-1}$ $y''=(1+x^2)^{-1}+(1+x^2)^{-1}+(-2x)(x+1)(1+x^2)'^{-2}$

9. amistre64

quotient rule is fine; i like to use product on negative exponents to make life a bit easier to compute

10. anonymous

i see

11. amistre64

to clean it up we: $y''=2(1+x^2)^{-1}+(-2x^2-2x)(1+x^2)^{-2}$ $y''=(2(1+x^2)+(-2x^2-2x))(1+x^2)^{-2}$ $y''=(2\ \cancel{+2x^2-2x^2}^{\ 0}-2x)(1+x^2)^{-2}$ $y''=(2-2x)(1+x^2)^{-2}$ y'' = 0 when x=1; and is not undefined for any values so our only option is at x=1 to "test out"

12. amistre64

0 : 0s and undefs <---------------------> - 1 + : curve directions

13. amistre64

well, I got me + and - on the wrong sides, but still; its the point of inflection :)

14. anonymous

so the answer is (-1, 0)?

15. anonymous

nooo i meant (0,1)???

16. amistre64

http://www.wolframalpha.com/input/?i=2nd+derivative+%28x%2B1%29arctanx i dropped a negative someplace but still the same basic results so inflection is x=1 (x=1,y=) .... need to plug in x=1 into the y= f(x) part to determine its value unless we only have 1 option

17. anonymous

i have the option there, so it's either d or e?

18. amistre64

correct, so when x=1 y=2 arctan(2)

19. anonymous

so (1, pi/2)?

20. amistre64

|dw:1328031733509:dw|

21. amistre64

i got no idea what angle that is :) wolf to the rescue

22. amistre64

arctan(1) right ....

23. anonymous

lol

24. anonymous

k

25. amistre64

2*45^ = 2*pi/4 = pi/2 yep

26. anonymous

thankss again :)

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