Find the points of local maxima or minima and corresponding local maximum and minimum values of `f(x)=x^5-5x^4+5x^3-1`

Explanation

`f(x) =x^(5) -5x^(4)+5x^(3)-1`
` f(x) =5x^(2) (x-1) (x-3)`
`f(x)=0 " "rArr " " x=0 ,1,3`
`f''(x) =10 x(2x^(2) -6x+3)`
`"Now " f''(1) lt 0 " "rArr " ""Maxima at " x=1`
`f''(3) gt 0 " "rArr " ""Minima at " x=3`
`"and " f''(0) =0 " "rArr " "II^(nd) " dervative test fails "`
`" So " f'''(x) =30 (2x^(2)-4x+1)`
`f'''(0) =30`
`rArr` Neither maxima nor minima at x=0
Note : It was very convenient to check maxima`//`minima at first step by examining the sigh change of f(x) no sigh change of f'(x) at x=0
`f'(x) =5x^(2)(x -1) (x-3)`