Let `f(x)={x^3-x^2+10 x-5,xlt=1, \ -2x+(log)_2(b^2-2),x >1` Find the values of `b` for which `f(x)` has the greatest value at `x=1.`

Answer

`b in [-sqrt(-130),-sqrt(2)]cup[sqrt(2),sqrt(130)]`

Explanation

for `xlt1 , f(x) =3x^(2)-2x+10gt0`
Thus f(X) is an increasing function for `xlt1`
For `xgt1, f(X) =-2`
Thus f(x) is a decreasing funciton for `xgt1` 1.Now f(X) will have greatest value at x =1 if
`underset(xrarr(1^(+))lim f(x)lef(1)`
or `-2+log_(2)(b^(2)-2)le5`
or `0ltb^(2)le128 or 2 le b^(2)le 130`
or `bin [-sqrt(-130),-sqrt(2)]cup[sqrt(2),sqrt(130)]`