Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5-5x4+5x3-1.

Find the points of local maxima local minima and the points of inflection of the function f(x) = x5-5x4+5x3-1. Also find the corresponding local maximum and local minimum values.

We have, f(x) = x,5, - 5x,4, + 5x,3, - 1,On differentiating w.r.t.x, we get ,f'(x) = 5x,4, - 20x,3, + 15x,2,For maxima or minima,f'(x) = 0,⇒ 5x,4, - 20x,3, + 15x,2, = 0,⇒ 5x,2,(x,2,-4x+3) = 0,⇒ 5x,2,(x-1)(x-3) = 0,⇒ x = 0,1,3,Sign scheme for f'(x) = 5x,2,(x-1)(x-3) is as shown in the following figure,So, y has maxima at x = 1 and minima at x = 3,At   x = 0, y has neither maxima nor minima, which is the point of inflection .,f(1) = 1-5+5-1=0,which is local maximum value at x = 1,f(3) = (3),5,-5(3),4,+5(3)3-1,= 243-81×5-27×5-1=-298, which is local minimum value at x = 3.