Find the points at which the function `f` given by `f(x)=(x-2)^4(x+1)^3` has local maxima local minima point of inflexion

Explanation

Given equation is ,
`f(x)=(x-2)^(4)(x+1)^(3)`
`f'(x) = (x-2)^(4)3(x+1)^(2)+(x+1)^(3)4(x-2)^(3)`
` = (x-2)^(3)(x+1)^(2){3(x-2)+4(x+1)}`
` = (x-2)^(3)(x+1)^(2)(7x-2)`
For maxima/minima, `f'(x)=0`
` rArr (x-2)^(3)(x+1)^(2)(7x-2)=0`
`rArr x= 2, -1, 2/7`
(i) At ` x = 2/7`,

`:. f'(x)` changes its sign from positive to negative, so ` x = 2/7` is a point of local maxima.
(ii) At x = 2,

`:. f'(x)` changes its sign from negative to positive, so x = 2 is a point of local minima.
(iii) At x =- 1,

`:. f'(x)` does not change its sign , so = - 1 is a point of inflexion.