EE 518 : Homework #5 Due on Monday 2:00pm, October 3, 2016Problem 1Suppose that f : (0, 1) → R is uniformly continuous on (0, 1). If {an} is a Cauchy sequence in (0, 1) andbn= f(an), show that {bn} is a Cauchy sequence in R.Problem 2True or false? Justify your answer. Give a counterexample if it is false.(a) If limx→af(x) = 0 and limx→ag(x) = 0, then limx→af(x)g (x)does not exist.(b) If neither limx→af(x) nor limx→ag(x) exists, then limx→a(f(x) + g(x)) does not exist.(c) If limx→af(x) exists, but limx→ag(x) does not exist, then limx→a(f(x) + g(x)) does not exist.(d) If limx→a(f(x)g(x)) exists, then the limit must be equal to f (a)g(a).(e) If limx→af(x) = ∞ and limx→ag(x) = ∞, then limx→a(f(x) − g(x)) = 0.Problem 3Find any local minimum or local maximum of following functions, if exist.(i)f(x) =12x2e−x(ii)f(x) =x1 + x2Problem 4(a) A critical point of a function f : X → R is a point c ∈ X such that either f0(c) = 0 or f0(c) does notexist. Consider the function f (x) = ax3+ bx2+ cx + d, where a 6= 0. Show that f can have two, one,or no critical points (give examples and sketches to illustrate the three possibilities). How many localextreme values can f have?(b) Show that the equation ex− 2 = cos√x has at least one root in (0, 1).(c) Consider the function f (x) = x4− 4x3+ 4x2+ c, where c is a constant. Show that f has at most onezero in the interval (1, 2). For what interval of values of c does the equation f(x) = 0 have exactly oneroot in (1, 2)?Problem 5Suppose function f has n derivatives on the interval (0, a) and limx→0+x2nf(n)(x) = L exists in R, wheref(n)denotes the n-th derivative of f. Find limx→0+xnf(x) in terms of L.Page 1 of 2EE 518 : Homework #5 Due on Monday 2:00pm, October 3, 2016Problem 6Given values of variables x and y:x = [0 0.3 0.8 1.1 1.6 2.2];y = [0.6 0.67 1.01 1.35 1.45 1.20];Fit the data with first, second, and fifth degree polynomials and plot all polynomial fits with different colorson the same figure. Submit your code along with your plot. Hint: use polyfit() and polyval()Page 2 of