If for all x such that |x – a| δ where f(x) ≤ f(a) [ or f(x) ≥ f(a)], f(a) is a relative maximam [ or relative minimum]. For f(x) to have a relative maximum or minimum at x = a, it must have f'(a) = 0. Then if f”(a) f”(a) ≥ 0 it is a relative minimum. Possible points at which f(x) has a relative maxima or minima are obtained by solving f'(x) = 0, i.e. by finding the values of x where the slope of the graph f(x) is equal to zero.
Similarly f(x, y) has a relative maximum or minimum at x = a, y = b if fx(a, b) = 0, fy(a, b) = 0. Thus possible points at which f(x, y) has relative maxima or minima are obtained by solving simultaneously the equations
Extensions to functions of more than two variables are similar.
