electrostatics - Minima & maxima of Laplace's equation

I don't get the following sentence from David J. Griffiths' Introduction to Electrodynamics (the ambiguous sentence is in bold)

Laplace's equation tolerates no local maxima or minima; extreme values of $V$ must occur at the end points. Actually, this is a consequence of (1), for if there were a local maximum, $V$ would be greater at that point than on either side, and therefore could not be the average. (Ordinarily, you expect the second derivative to be negative at a maximum and positive at a minimum. Since Laplace's equation requires, on the contrary, that the sec­ond derivative is zero, it seems reasonable that solutions should exhibit no extrema. However, this is not a proof, since there exist functions that have maxima and minima at points where the second derivative vanishes: $x^4$ for example, has such a minimum at the point $x = 0$.)