Maxima and Minima

If for all x such that |x - a| < δ we have f(x) ≤ f(a) (or f(x) ≥ f(a)), we say that f(a) is a relative maximum (or relative minimum). For f(x) to have a relative maximum or minimum at x = a, we must have f'(a) = 0. Then if f"(a) < 0 it is a relative maximum while if f"(a) > 0 it is a relative minimum. Note that possible points at which f(x) has a relative maxima or minima are obtained by solving f'(x) = 0, i.e. by finding the value of x where the slope of the graph of f(x) is equal 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

δf/δx = 0, δf/δy = 0

Extensions to function of more than two variables are similar.

Method of Langrange Multipliers
Sometimes we wish to find the relative maxima or minima fo f(x,y) = 0 subject to some constraint condition φ(x,y) = 0. To do this we form the function h(x,y) = f(x,y) + λφ(x,y) and set

δh/δx = 0, δh/δy = 0

The constant λ is called a Langrange multiplier and the method is called the method of Langrange multipliers. Generalizations can be made.

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