Find the maximum and minimum values, if any, of the following functions given by (i) `f" "(x)" "=" "|x" "+" "2" "|" "" "1` (ii) `g(x)" "=" "" "|" "x" "+" "1|" "+" "3 ` (iii) `h(x)" "=" "s in" "(2x)" "+" "5` (iv) `f" "(x)" "=" "|" "s in" "4x" "+"

Explanation

(i) `f(x) = |x+2|-1`
For each `x in R`
`|x+2| ge 0`
` rArr |x+2|-1 ge 0-1`
` rArr f(x) ge -1`
`:.` minimum value of f(x) =-1
It has no maximum value.
(ii) ` g(x) =- |x+1|+3`
For each `x in R`
`|x+1| ge 0`
` rArr -|x+1| le 0`
` rArr -|x+1|+3 le 0+3`
` rArr g(x) le 3`
`:. ` maximum value of g(x) is 3.
It has no minimum value.
(iii) ` h(x) = sin (2x)+5`
We know that
` -1 le sin (2x) le 1`
` rArr -1 + 5 sin (2x)+5 le 1 + 5`
` rArr 4 le h(x) le 6`
`:.` minimum value of h(x) is 4 and maximum value is 6.
(iv) ` f(x) = |sin 4x + 3|`
We know that
` -1 le sin 4x le 1`
` rArr -1 + 3 le sin 4x + 3 le 1+3`
` rArr 2 le sin 4x + 3 le 4`
` rArr 2 le |sin 4x + 3| le 4`
` rArr 2 le f(x) le 4`
`:.` minimum value of f(x) is 2 and maximum value is 4.
(v) In `x in (-1, 1), h(x) = x+1`
` :. -1 lt x lt 1`
` rArr -1 + 1 lt x + 1 lt 1 + 1`
` rArr 0 lt h(x) lt 2`
` :.` h(x) has neither a minimum value nor a maximum value.