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.

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$