f at x = – and – are f ( ) = and f (– ) = – , respectively. Example Find all the points of local maxima and local minima of the function f given by f ( x ) = x – x + x + . Solution We have f ( x ) = x – x + x + ( ) ( ) ′ ′′ Now f ′ ( x ) = gives x = . Also f ″ ( ) = .
Therefore, the second derivative test fails in this case. So, we shall go back to the first derivative test. We have already seen (Example ) that, using first derivative test, x = is neither a point of local maxima nor a point of local minima and so it is a point of inflexion. Example Find two positive numbers whose sum is and the sum of whose squares is minimum.
Solution Let one of the numbers be x . Then the other number is ( – x ). Let S( x ) denote the sum of the squares of these numbers. Then S( x ) = x + ( – x ) = x – x + S ( ) S ( ) ′ ′′ Now S ′ ( x ) = gives x = .
Also S ′′ > . Therefore, by second derivative test, x = is the point of local minima of S. Hence the sum of squares of numbers is minimum when the numbers are and Remark Proceeding as in Example one may prove that the two positive numbers, whose sum is k and the sum of whose squares is minimum, are and k k . Example Find the shortest distance of the point ( , c ) from the parabola y = x , where ≤ c ≤ .