How to derive all segments above a given function

pcqsl

Hope I am asking question at the right board.

I have a differentiable real function f(x).  I am looking for a linear functioin t(x), in the form t(x)=s*x, such that t(x)>=f(x) within an interval (a,b).

Is there a systematic way to have all solutions of s for any interval (a,b)?

molen

Well, interetsing problem.
I guess you can construct your t(x) with the three values: f(a), f(b), and max(f) when f is inside the interval (a,b).

jiguixiu

how about dividing interval (a,b) into many small intervals,and in any small interval  t(x) = max(f). not in the  form t(x)=s*x, isn't?

pcqsl

原帖由 jiguixiu 于 2008-3-4 09:34 发表
how about dividing interval (a,b) into many small intervals,and in any small interval  t(x) = max(f). not in the  form t(x)=s*x, isn't?

Thank you for helping.  But the form t(x)=s*x is what I really want.

Let us consider a simpler problem.  We assume f(x) is also bounded, and for a given interval (a,b) , such that a>0.  It is easy to see that if s1 is a solution, then for any s>s1 is a solution, too.  In other words, there exists s0, which is the minimum or infimum of all solutions.  Now, my question is how to get this value?

And I think f(a) , f(b) and max_{x\in(a,b)} f(x) may not enough to determine such s0.  I guess the first order derivative involves, as well as the second order derivative.

pcqsl

原帖由 molen 于 2008-3-4 08:46 发表
Well, interetsing problem.
I guess you can construct your t(x) with the three values: f(a), f(b), and max(f) when f is inside the interval (a,b).

Thank you.  You may give me more hints for the following explicit example.

Let us consider a simplest case: f(x)=3-(x-3)^2.  In the interval (1.5, 3), we can find s0=6-2*6^(0.5), such that for all s>=s0, t(x)=s*x >= f(x).  But f(a), f(b), and  max(f)=f(b), are not sufficient to determine s0.

molen

Okay, the problem is that you did not give us enough clue for your problem.

In the example, you specify that the "linear function" t(x) must pass the origin. (you did not tell us this constraint).
Based on your above example, this is a very simple problem. If f(x) is continuous and differentiable, then all you need is to find tha tangent line from origin.

If f(x) is not a differentiable function, what you need is another kind of "optimization problem".
Considering the case that both a and b are located at the right of origin (if (a,b) includes origin, than you can not get the linear function t(x) you want), what you need is to rotate the line OA which is initially along x-axis, the line OA with a maximal angle theta w.r.t x-axis, and pass through f(x) is what you want.

An I right?

pcqsl

Thanks, molen. Intuitively, you are right.  I also realize that the key is the tangent line.

By the way, you claim that: if (a,b) includes origin, than you can not get the linear function t(x) you want.  This may not be always true.  Consider my last example, if I just change (a,b) from (1.5,3) to (-3,3), the solutions have the exact same lower bound as before.  Plus, there is a upper bound on s, too.  Of course, if a function f(x) , such that f(0)>0, there is no solution at all.

My problem is that I don't have an explicit function f(x).  I hope the result can apply to a family of functions with  certain properties, i.e. differentiable.   I  would like to keep things in abstract manna and extract the sufficient conditions. But I do not want such conditions to be too restricted.   Using the above example again, tangent line is the solution because that function f(x) is a global concave function. Otherwise, for a convex function, for example, the tangent line is blow the function.  So, at this moment, we may simply say: for global concave differentiable functions, tangent lines will be the key for solutions.  But "global concave" is a too strong limitation, even it is concave within the interval (a,b) is too restricted.  I must consider more general functions.  On the other hand, it is hard to tell if a condition is too restricted or not before it is given.