. If x, y E R with x < y, prove that x < tx + (1 – t)y < y for all t e (0,1).

4. Let u be a root of ƒ = t³ − t² + t + 2 € Q[t] and K = Q(u). (a) Show that f = mo(u). (b) Express (u²+u+1) (u²-u) and (u-1)-¹ in the form au²+bu+c, for some a, b, c € Q.

Find the condition of c such that x² + 2y² +3z² + c(xy + yz) > 0 for all x, y, z & R with x² + y² + z² # 0.

6. If af/dx is continuous on the rectangle R = {(t, æ) : 0 0 such that |f(t, x1) – f(t, x2)| < K\x1 – x2| for all (t, æ1) and (t, x2) in R. 7. Define the sequence {un} by uo (t) = 20, Un+1 = x0 + to + f(s, un(8)) ds, п %3 1,2,.... Use the result of the previous exercise to show that |f(t, un (t)) – f(t, un-1 (t))| < K|u,(t) – Un-1(t)|.