Methods of Finding Maxima and Minima
There are two methods of finding the maximum and minimum values of a function.
They are: (1) Method of first order of differentiation and (2) Method of higher order of differentiation.
Both the above methods are briefly depicted here as under:
1. Method of First order of differentiation.
Under this method, the following steps are taken up in turn to find the maximum and minimum values of a function of one variable.
Step 1. Find dy/dx fir the given function.
Step 2. Determine the values of x which make dy/dx = 0
Step 3. Test the values thus obtained to ascertain, if they give maximum or minimum value. For this, study the sign of dy/dx for the value say a, by trying with slightly < a and slightly > a. If dy/dx for the value, a changes sign from + ve to – ve, then the f(x) has a maximum value at x = a, and the max f(x) = f(a). On the other hand, if dy/dx for the value a changes sign from – ve to + ve, then the f(x) has a minimum value at x =a and the min f(x) = f(a).
If dy/dx does not change sign then, as stated earlier, x = a is a point of inflexion. From the practical point of view, the method elucidated above however, is a tedious one as it needs a careful examination of the change of signs from one direction to another as required under the step 3. Hence, this method should be avoided unless specifically asked for.
2. Method of Higher order differentiation.
Under this method, the following steps are taken up in turn to arrive at the maximum and minimum values of a function of one variable.
Step 1. Find dy/dx for the given function.
Step 2. Determine the values of x which make dy/dx = 0
Step 3. Find d²y/dx² and evaluate the same straight way, or by putting x = in it if x exists. If d²y/dx² gives zero, then either follow the first order method explained above, or find out ht next higher order differentiation successively viz. d²y/dx², d4y/dx4
etc. by putting x = a there in till they yield some value other than zero. From practical point of view, the working may be continued up to the 4th order, and if the same, even, yields zero, note that the point x =a, is a point of inflexion, where the function remains stationary without any change.
If d²y/dx², or any of its succeeding derivatives gives a – ve value, the function is said to be at a maximum at x =a. On the other hand, if any of the above higher order derivatives gives a +ve value, the function is said to be a t a minimum at x = a.
