Find the points of local maxima, local minima and the points of inflection of the function `f(x)=x^5-5x^4+5x^3-1` . Also, find the corresponding local maximum and local minimum values

Explanation

Given that, `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
`rArr 5x^(4)-20x^(3)+15x^(2)`
`rArr 5x^(2)(x^(2)-4x+3)=0`
`rArr 5x^(2)(x^(2)-3x-x+3)=0`
`rArr 5x^(2)[x(x-3)-1(x-3)]=0`
`therefore x=0,1,3`
Sign scheme for `(dy)/(dx) = 5x^(2)(x-1)(x-3)`

So, y has maximum value at x=1 and minimum value at x=3.
At x=0, y has neither maximum nor minimum value.
`therefore` Maximum value of y=1`-5+5-1=0`
and minimum value `=(3)^(2)-5(3)^(4)+5^(3)-1`
`=243-81 xx 5-27 xx 5-1=-208`