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Suppose I have a function V(x,y,z) which obeys Laplace's equation over some path in space. That is to say, for some path parametrized by [itex]\vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}[/itex], [itex]a\leq t \leq b[/itex], [itex]\nabla ^2 V(\vec{r}(t))=0[/itex].

Is it true that the extreme values of V along that path are located at the end? (I.e. at [itex]V(\vec{r}(a))[/itex] and [itex]V(\vec{r}(b))[/itex])?