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The greatest bending moment in this setup sits at the center of the beam. Here’s why: the left support reaction is P/2, so on the left half the internal moment grows linearly from zero at the left end up toward a peak as you approach the middle. The central load then causes the shear to jump, and on the right half the shear becomes negative, so the moment there decreases linearly from that same peak down to zero at the right end. By symmetry, the peak value occurs right at the midpoint. The maximum moment magnitude is Mmax = P times the span length divided by four, so it peaks in the middle rather than at the supports or elsewhere along the beam.