EE 518 Homework 5 Due on Monday 2 00pm October 3 2016 Problem 1 Suppose that f 0 1 R is uniformly continuous on 0 1 If an is a Cauchy sequence in 0 1 and bn f an show that bn is a Cauchy sequence in R Problem 2 True or false Justify your answer Give a counterexample if it is false a If limx a f x 0 and limx a g x 0 then limx a f x g x does not exist b If neither limx a f x nor limx a g x exists then limx a f x g x does not exist c If limx a f x exists but limx a g 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 a f x and limx a g x then limx a f x g x 0 Problem 3 Find any local minimum or local maximum of following functions if exist i f x 1 2 x x e 2 f x x 1 x2 ii Problem 4 a A critical point of a function f X R is a point c X such that either f 0 c 0 or f 0 c does not exist 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 local extreme 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 one zero in the interval 1 2 For what interval of values of c does the equation f x 0 have exactly one root in 1 2 Problem 5 Suppose function f has n derivatives on the interval 0 a and limx 0 x2n f n x L exists in R where f n denotes the n th derivative of f Find limx 0 xn f x in terms of L Page 1 of 2 EE 518 Homework 5 Due on Monday 2 00pm October 3 2016 Problem 6 Given 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 colors on the same figure Submit your code along with your plot Hint use polyfit and polyval Page 2 of 2