Find the intervals in which the following functions are strictly increasing or decreasing: (a `x^2+2x-5` (b) `10-6x-2x^2` (c) `-2x^3-9x^2-12 x+1` (d) `6-9x-x^2` (e) `(x+1)^3(x-3)^3`

Explanation

(a) `f(x) = x^(2)+2x+5`
`f'(x) = 2x+2`
Let` f'(x) = 0`

`rArr 2x+2=0`
` rArr x=-1`

`:. F(x)" is strictly decreasing in "(-infty, -1)" and strictly increasing in "(=1, infty)`.
(b) `f(x) = 10 - 6x - 2x^(2)`
` rArr f'(x) = - 6 - 4x`
Let `f'(x) = 0`
` rArr -6 - 4x = 0`
` rArr x=- 3/2`


`:. f(x)" is strictly decreasing in "(-infty, -3/2)" and strictly increasing in "(-3/2, infty)`.
(c) `f(x) = - 2x^(3)-9x^(2)-12x - 1`
`rArr f'(x) =-6x^(2)- 18x - 12`
` =-6(x^(2)+3x+2)`
` =-6(x+1)(x+2)`

Let ` f'(x) = 0`
` rArr -6(x+1)(x+2)=0`
`rArr x+1=0 or x+2= 0`
` rArr x=-1 or x=-2`

`:. f(x)" is strictly decreasing in "(-2, -1) " and atrictly increasing in "(-infty, -2) cup (-1, infty)`.
(d) ` f(x) = 6 - 9x-x^(2)`
Let `f'(x) = 0`
` rArr -9-2x = 0`
`rArr x= (-9)/2`
`:. f(x),"is strictly decreasing in "(-infty,(-9)/2)" and strictly increasing in"((-9)/2, infty)`.
(e) ` f(x) = (x+1)^(3)(x-3)^(3)`
` rArr f'(x) = (x+1)^(3)*3(x-3)^(2)+(x-3)^(3)*3(x+1)^(2)`
` = 3(x+1)^(2)(x-3)^(2)(x+1+x-3)`
` = 6(x+1)^(2)(x-3)^(2)(x-1)`
If `f'(x) gt 0`
then ` 6(x+1)^(2)(x-3)^(2)(x-1) gt 0, x ne - 1, x ne 3`
`rArr x-1 gt 0`
` rArr x gt 1`
`:. f(x)` is strictly increasing in the interval ` (1, infty) - {3}`.
Again `f'(x) lt 0`
`rArr 3(x+1)^(2)(x-3)^2) (x-1) lt 0 x ne - 1, x ne 3`
`rArr x- 1 lt 0`
` rArr x lt 1`
`:. f(x)" is strictly decreasing in the interval "(-infty, 1)-{-1}`.