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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
2 months, 4 weeks ago

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