If function and its derivative are zero at a point, then is the second and third derivative also zero [fourth order differential]?

Asked Oct 24 '22 at 18:42

Active Oct 24 '22 at 18:53

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The example I am working on is that of a string attached and deflected by a force along its length. Edit: There is something supporting it in the middle. We can say that the deflection, $f(x)$, must be zero at the frontier $f(0)=f(L)=0$ along with its derivative $\frac{\partial f(0)}{\partial x}=\frac{\partial f(L)}{\partial x} = 0$

Does the second and third derivative also evaluate zero at $x=0$ and $x=L$, considering that it is a fourth order differential equation?

Edit: the deflection can be expressed as $\frac{\partial^4 f(x)}{\partial x^4} = c P(x)$ with $P(x)$ being the pressure applied

edited Oct 24 '22 at 18:53

asked Oct 24 '22 at 18:42

Mooder

1

$f(x)=x^2(x-L)^2$ is an obvious counterexample. – mathcounterexamples.net Oct 24 '22 at 18:44
@mathcounterexamples.net: But doesn't OP require zero first derivatives at two points also? Wouldn't we need something like $\cos \frac{2\pi x}{L} - 1$ or something? ETA: OBE. – Brian Tung Oct 24 '22 at 18:46
@mathcounterexamples.net, it is a fourth order differential equation. – Mooder Oct 24 '22 at 18:52
@BrianTung I updated the comment with an example fulfilling that constraints. – mathcounterexamples.net Oct 24 '22 at 18:52
@Mooder Th OP updated the question... Always nice to shoot at a moving target! – mathcounterexamples.net Oct 24 '22 at 18:58
@mathcounterexamples.net Yes I saw that. – Brian Tung Oct 24 '22 at 21:52

If function and its derivative are zero at a point, then is the second and third derivative also zero [fourth order differential]?

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