then f ( c ) is a local minimum value of f . Figures . and . , geometrically explain Theorem .
Fig . Example Find all points of local maxima and local minima of the function f given by f ( x ) = x – x + . Solution We have f ( x ) = x – x + f ′ ( x ) = x – = ( x – ) ( x + ) f ′ ( x ) = at x = and x = – Thus, x = ± are the only critical points which could possibly be the points of local maxima and/or local minima of f . Let us first examine the point x = .
Note that for values close to and to the right of , f ′ ( x ) > and for values close to and to the left of , f ′ ( x ) < . Therefore, by first derivative test, x = is a point of local minima and local minimum value is f ( ) = . In the case of x = – , note that f ′ ( x ) > , for values close to and to the left of – and f ′ ( x ) < , for values close to and to the right of – . Therefore, by first derivative test, x = – is a point of local maxima and local maximum value is f (– ) = .
Values of x Sign of f ′′′′′ ( x ) = ( x – ) ( x + ) Close to to the right (say . etc.) > to the left (say . etc.) < Close to – to the right (say . etc.) to the left (say .
etc.) > Fig . Example Find all the points of local maxima and local minima of the function f given by f ( x ) = x –