## anonymous 3 years ago find f $f''(x) = x^{-2}, \quad x > 0,\quad f(1) = 0, \quad f(2) = 0$

1. myininaya

To find f you need to first find f' the find f' given f'' integrate both sides

2. anonymous

hmm.. $f'(x) = \frac 1x + c$ ??

3. myininaya

almost

4. myininaya

you are missing a certain constant multiple

5. anonymous

ahh $f'(x) = -\frac 1x + c$

6. myininaya

yes :) now to find the constant hmmm....you are missing a certain initial condition to do that .... your question doesn't make sense .... you need one of those to be f'(something)=another something

7. anonymous

or maybe i should do it $f'(x) = -\frac 1x + c_1$

8. anonymous

i might have typoed

9. Zarkon

integrate again

10. anonymous

ah yes i did. it's f'(2) not just f(2)

11. Zarkon

get a system of 2 eau and 2 unknowns

12. anonymous

hmm $f(x) = -\ln x + c_1 x + c_2$ yes?

13. myininaya

ok great. you can find that constant by using f(1)=0 and then do what zarkon says to find f

14. Zarkon

you can do the problem with two given values of f

15. myininaya

or you can wait to find the first constant whatever

16. anonymous

i suppose x > 0 is just there to note that -ln x exists?

17. myininaya

oh wait.... i guess you can do it with f(something1)=another something1 and f(something2)=another something2 oops

18. anonymous

f'(2) = 0 so f'(2) = -1/2 + c_1 = 0 does this mean c_1 is 1/2?

19. myininaya

yes adding 1/2 to both sides solves that equation for c_1

20. anonymous

then f(1) = -ln (1) + 1/2 x + c_2 = 0 so c_2 is -1/2?

21. myininaya

x is 1 so you have -ln(1)+1/2(1)+c_2=0 and yes

22. anonymous

oh. yeah...forgot to sub 1 into x the second time

23. anonymous

so $f(x) = -\ln x + \frac 12 x - \frac 12$ ??

24. myininaya

tep

25. anonymous

nice. thanks

26. anonymous

i just noticed this was my 1000th question