Let f(x) `underset( 2X -3 " "1 lt X lt 3)(={ -X^(2) +((b^(3) -b^(2) +b-1))/((b^(2) +3b+2))}} 0le X lt1`
Find al possible values of b such that f(x) has the smallest values at x=1

Explanation

Such problems can easily be solved by graphical approach (as in figure).

Hence the limiting value of f(x) from of x=1 should b either greater of equal to the value of function at x=1.
` underset( x to1)( "lim") f(x) le f(1)`
`rArr " " - 1 + ((b^(3) -b^(2) +b-1))/((b^(2)+3b+2)) le -1`
`rArr " " ((b^(2) +1)(b-1))/((b+1)(b+2)) ge 0`
`rArr" " b in (-2 ,-1) uu [1,+oo)`