In the book titled Classical Theory of Fields, by famous Physicist Landau, writes,
From $\nabla^2\phi(x,y,z)=0$,
...it follows, in particular, that the potential $\phi$ of the electric field can nowhere have a maximum or a minimum. For in order that $\phi$ have an extreme value, it would be necessary that the first derivatives of $\phi$ w.r.t the coordinates be zero, and that the second derivatives, $\partial^2\phi/\partial x^2,\partial^2\phi/\partial y^2,\partial^2\phi/\partial z^2$ all have the same sign. The last is impossible since the Laplace's equation cannot be satisfied.
Is it true that the second derivatives $\partial^2\phi/\partial x^2,\partial^2\phi/\partial y^2,\partial^2\phi/\partial z^2$ must necessarily have the same sign at a local extremum i.e., all negative for local maximum and all positive for a local minimum? Thinking geometrically, probably the answer is yes. But I am not sure.
The argument is not correct. Even at a strict local extremum, all second partial derivatives can be zero; consider for example $\phi(x,y,z) = x^4+y^4+z^4$ at the point $(0,0,0)$.
(Correct statements and proofs of the weak/strong maximum principles can be found in many PDE textbooks.)