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Vertical displacement of a curve

Lets say we have an unknown function \(y=f(x)\) valid between the domain \(x=a\) to \(x=b\). Now the unknown function is continuous throughout the whole domain \(a\) to \(b\) and obtains a maximum between the points a to b at some certain value of \(x\).

We do not have any other information apart from this and hence this curve cannot be uniquely determined theoretically.

Now if the point \(a\) and \(b\), which are the endpoints of the curve get vertically displaced by amount A and B respectively (A and B need not be very large, in fact, they are small displacements only) can we say in a hand waving way the net vertical displacement of the curve will be proportional to the relative difference between A and B?

It need not be exact, I understand but at least to some order of accuracy can this statement hold true?

It is related to some physical problem on turbulence which I am working, so would very much appreciate the response from you guys.

Note by Subharthi Chowdhuri
11 months, 3 weeks ago

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