need proof for simple geometry problem
- From: "mbekkali" <mbekkali@xxxxxxxxx>
- Date: 30 May 2005 14:07:07 -0700
Good day. Here is my problem.
Assume one has a continious and twice (or more) differentiable function
function g[x] such that g'[x]<0 and g''[x]>0, where x belongs to [0,1].
At any point x0 one can find the ratio of the area of an integral of
g[x] from [0,x0], call it G[0,x0] to the sum of integrals G[0,x0] and
G[x0,1]. Let's call this ratio m (again, m=G[0,x0]/(G[0,x0]+G[x0,1])).
Further, at any point x0 one can draw a tangent line to g[x], call it
t[x,x0]. Let's call the area below the tangent line from [0,x0] as
T[0,x0]. Then, similarly we can define n=T[0,x0]/(T[0,x0]+T[x0,1]). I
need to show that n/m is finite for any point x0 as long as g[x]
satisfies the assumptions laid out above.
I used different specific functional forms of g[x] that satisfy the
initial assumptions and what I found is that n/m<3/2. However, I have
not had luck proving it analytically. If you can derive this magic
upper threshold and let me know what it is, that woud be great. Thanks,
Mukhtar Bekkali
.
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