Worked example: A particle in a region has ψ(x) = A·x·(L − x) for 0 < x < L (and zero elsewhere). Where is the particle most likely to be found? Step 1: Probability density P(x) = |ψ|² = A²x²(L − x)². Step 2: Maximise: dP/dx = 0. Write P ∝ x²(L−x)²; d/dx gives 2x(L−x)² − 2x²(L−x) = 2x(L−x)(L − 2x) = 0. Step 3: Solutions: x = 0, x = L (endpoints, minima) and x = L/2 (maximum). Step 4: So the most likely position is the centre x = L/2, and the particle is never found at the walls (nodes). Conclusion: the shape of |ψ|² directly answers 'where will I find it?' — probability peaks where the wave function bulges.