## #math - Sun 1 Apr 2007 between 01:35 and 01:42

### NY Lost Funds

 TRWBW mingkus: um, nopemindwarp: it's f(x)=log(1+x), f'(x)=1/(1+x) futurist 0 and ln2 TRWBW futurist: f'(x) not f(x) futurist oh, 1/2 and 1 mingkus you are trying to prove this right x/2 < log(1+x) < x ?? TRWBW futurist: and if using the mean value you get x*f'(t) for some t in [0,x] which means t also in [0,1], what can you say about x*f'(t)?futurist: if f'(t) is in [1/2,1], what can you say x*f'(t) is in? futurist lemme think about that for a second TRWBW futurist: if a is in [1/2,1], what can you say a*b is in? futurist oh [b/2,b] -- i was starting at the x*f'(t) thing mingkus futurist: x/2 < log(1+x) < x for x in [0,1], is that what you want to prove? TRWBW futurist: okay, so back to the last onefuturist: if f'(t) is in [1/2,1], what can you say x*f'(t) is in? futurist mingkus yeahoh, [x/2,x]hmm weird TRWBW futurist: look helpful? mingkus futurist: is it log(1 + x) or ln(1 + x) Safrole How about using the convexity of the log function?Is that applicable to this problem? futurist so x/(1+x) is in the desired intervale TRWBW ? futurist f'(x) = 1/(1+x) mingkus not if it log futurist i mean, x/(1+x) is in the interval we want to show log(1+x) is inln TRWBW futurist: you want to show that f(x) is in [x/2,x]. MVT tells you f(x)=f'(t)*x for some t in [0,x]. you just showed f'(t)*x is in [x/2,x] futurist log = ln in this book mingkus log at base e is equal to lnk i know how to do it futurist mingkus yes action TRWBW shaves his head TRWBW shaves his headoops shakes mingkus devide in 2... Safrole lol mingkus x/2 < log(1+x) and log(1+x) < x TRWBW futurist: you want to show that f(x) is in [x/2,x]. MVT tells you f(x)=f'(t)*x for some t in [0,x]. you just showed f'(t)*x is in [x/2,x] mingkus or ln sorry futurist ah