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

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