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EXERCISES, , . Show that a continuous function on a closed bounded interval is bounded. Give an, example to show that conclusion fails if the interval is not closed., [Delhi Maths (G), 1996], . Let f be continuous on [a, b]. Prove that fattains its supremum and infimum on [a, 5]., [Delhi Maths (G), 1999, 2000], . (a) Ifa function fis continuous in [a, b] and c € [a, 5] such that f(c) # 0, then there, exists some 5 > 0 such that f(x) has the same sign as f(c) for all x € Je— 8, c + OI., [Delhi B.A. (Prog.) III, 2013; Meerut, 1998], (b) Let f be continuous on [a, b] and x, be such that a < xq <b. If f(xo) < 0, then show, that f(x) <0 V x © Jxy — 5p, Xp + Sol for some 5 > 0., [Delhi Maths (G), 1993], . If a function f is continuous in [a, 5] and f(a) f(b) < 0, then there exists a point, c € [a, b] such that f(c) = 0., Or, If a function fis continuous in f(a) > 0 > f(b), then there exists a point c € [a, b] such, that f(c) = 0., [Delhi Maths (Hons), 1997; Delhi Physics (Hons), 1996; Delhi Maths (G), 1994], , . Let fbe continuous on [a, b] and c be any real number between f(a) and f(b), then show, that there exists a real number x in Ja, b[ such that f(x) = c. [Delhi Maths (H), 2002], , . Let fbe continuous on [a, 5] and Xj, Xp5 -+9 X, be points of [a, b]. Show that there exists, apoint c € [a, b] such that f(c) =[f(x)) +f) + ----. +f (x,)\/n., , [Delhi Maths (G), 1999], log (242) 27" sin x, , ~~ Let = li koe 0,, £@) n re eae, Show that (0) and _f(n/2) differ in sign. Why does f(x) not vanish in [0, 2/2] ? Explain., : Bn+2 = Gogh:, . Show that the function f(x) = lim ies ee does not vanish anywhere in the, n>o x" +4, , interval [0, 2], though (0) and f(2) differ in sign., , 1