Linear programming & optimisation

Linear programming is optimisation with straight lines: maximise or minimise a linear objective subject to linear constraints, with all variables non-negative. The key theorem is that the optimum always occurs at a corner point (vertex) of the feasible region, so the method is mechanical: draw the region, list its corners, evaluate the objective at each, and pick the best. Worked example: maximise 3x + 4y subject to x + y <= 10, x <= 6, y <= 7, x, y >= 0. The corners are (0, 0), (6, 0), (0, 7), (6, 4) and (3, 7). Evaluating gives 0, 18, 28, 34 and 37, so the maximum is 37 at (3, 7). No calculus is needed — corners decide everything.