Number of integral values of `b` for which the equation `(x^3)/3-x=b` has three distinct solutions is____

Answer

1

Explanation

`f(x)=(x^(3))/(3)-x-b`
`therefore f(x)=x^(2)-01=0`
`therefore f(X)=x^(2)-1=0`
`therefore x=1 or -1`
for three distinct roots `f(x_(1)).f(x_(2))lt0` where `x_(1)` and `x_(2)` are the roots of f(x) =0 thus
`(1/3-1-b)(-1/3+1-b)lt0`
or `(b+2/3)(b-2/3)lt0`
or `b in (-2/3,2/3)`