Mathematically, we can see that with downward loads only (only upward loads at the ends), the maximum bending moment for a fixed end beam of span $L$ either occurs at $x=0$, $x=L$, or where the shear force is zero (or discontinuous, or otherwise undefined).
Is is the case that the maximum (absolute value) bending moment is in fact always found at one of the ends?
The most extreme case I can envision is a point load at the midpoint of the beam, this gives the (absolute value) bending moment equal, and maximised, at the ends and the midpoint: $$M(0)=M(L/2)=M(L).$$
I have asked this in the mathematical context here. I have a semi-rigorous proof there.