2. Show that the following functions are convex by verifying the condition that V² f(x) 20 is satisfied for all a in the domain of f: (a) f(un, u₂) In(e" + e"), (b) f(u, uz, us, UA) -In(1-₁-2-3-4) over the domain {u E R¹|u₁+uz+us+u₁ < 1}.

Exercise II.5. (a) Which are the following functions (defined from R to R) are convex: (i) f(x) = e (iii) f(x) = (x² - 1)² (iv) fs = ² (b) Show each function in Exercise II.1(a) is convex. (i) fi I1 X₂ (ii) fe(x) = 2x²-3x+5 I1 12 (ii) f2 (11) := x² + x² - 2 (iii) f3 := 9(x₁ − 1)² + (2+1)²-2 (iv) f₁ = √√√(x₁ + 1)² + (x₂ − 2)² - | := 4x² + ² − 1 x1 In

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.

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