Find all possible values ( range) of the following quadratic expressions when `x in R` and when `x in [-3,2]`
(a) `4x^2+28x+41`
(b) `1+6x -x^2`

Answer

(a) `[-8 , oo ) " when " x in R;[-7 113] "when" x in [-3 , 2 ] `
(b) `(-oo , 10 ] " when " x in R ; [ -26 , 9 ] " when " x in [-3,2] `

Explanation

`4x^2+28 x+41 =(2x +7)^2-8`
Now `(2x+7)^-8`
`rArr (2x+7)^2-8 ge -8 AA x in R`
So, the values of the expression are` [-8,oo)`
For `x in [-3,2]`
` -3 le x le 2 `
`rArr -6 le 2x le 4`
`rArr 1 le 2x +7 le 11`
`rArr 1 le (2x +7 )^2 le 121`
`rArr -7 le (2x+7)^2-8 le 113`
Thus for `x in [-3 , 2]` the values of the expression are [-7 ,113]
(b) `1=6 x-x^2 =10 -(x-3)^2-8` ltbr gt `Now (x-3)^2 ge 0 AA x in R`
`rArr -(x-3)^2 le 0 AA x in R`
`rArr 10 - (x-3)^2le 10 -(x-3)^2 le 10 AA x in R `
So, the values of the expression are `(-oo,10)`
For `x in [-3 ,2]`
`-3 le x le 2`
`rArr -6 le (x -3 )^2 le 36`
`-36 le - (x -3)^2 le -1`
`1 le (x -3)^2 le -(x - 3)^2 le -1`
`rArr -26 le 10 -(x -3)^2 le 9`
Thus ,for `x in [-3 , 2]` values of the expression are [ -26, 9]