Solve the below:
Q: Suppose that f: [a,b]→ R is differentiable, that 0 < m f ‘(x) M for x ? [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 ? [a,b], the sequence (xn), xn+1 = xn -1/M for n = 1, 2,..., is well defined (i.e. for each n, xn ? [a,b]), and that xn ∈(a,b)]and that xn→x‾ as n →∞, where ? [a,b] is such that f( ) =0.