Find the maximum and minimum values, if any, of the following functions given by(i) `f(x)=(2x-1)^2+3` (ii) `f(x)=9x^2+12 x+2`(iii) `f(x)=-(x-1)^2+10` (iv) `g(x)=x^3+1`

Explanation

(i) `f(x) = (2x-1)^(2)+3`
For each `x in R, (2x-1)^(2) ge 0`
`rArr (2x-1)^(2) + 3 ge 0+3`
`rArr f(x) ge 3`
`:.` minimum value of f(x) = 3.
It has no maximum value.
(ii) `f(x) = 9x^(2) + 12x +2`
` = (9x^(2)+12x+4) -2`
`=(3x+2)^(2)-2`
For each `x in R, (3x + 2)^(2) ge 0`
` rArr (3x + 2)^(2) - 2 ge 0 - 2`
` rArr f(x) ge - 2`
`:.` minimum value of f(x) =-2
It has so maximum value.
(iii) `f(x) = - (x-1)^(2) + 10`
For each ` x in R, (x-1)^(2) ge 0 `
`rArr -(x-1)^(2) le 0`
` rArr -(x - 1)^(2) + 10 le 10`
` rArr f(x) le 10`
`:. f(x)` has maximum value 10.
It has no minimum value.
(iv) ` g(x) = x^(3)+1`
As x increases, g(x) also increases. So g(x) has no maximum value.
Similarly as x decreases, g(x) also decrease.
So, g(x) has no minimum value.