Let a and b be arbitrary real numbers with a < b. (a) Let h(r) be...

 

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Let a and b be arbitrary real numbers with a < b. (a) Let h(r) be a continuous function on [a, b], and let m be a positive even integer. Suppose that ["h(z) dr = 0. Prove that h(r) is identically zero (meaning that h(r) - 0 for every re [a,b]). (b) Does there exist a continuous function f(r) on 1-1, 1] that is not identically zero such that f(r)" dr = 0 for every positive odd integer n? (c) Does there exist a continuous function j(r) on [a, b] that is not identically zero such that j(x)" dr = 0 for every positive odd integer n? (Hint: adapt your solution to part (b).) (d) Let k(r) be a continuous function on [a, b]. Suppose that for every continuous function () on [a, b], we have [k(r)f(x) dr = 0. Prove that k(r) is identically zero. Let a and b be arbitrary real numbers with a < b. (a) Let h(r) be a continuous function on [a, b], and let m be a positive even integer. Suppose that ["h(z) dr = 0. Prove that h(r) is identically zero (meaning that h(r) - 0 for every re [a,b]). (b) Does there exist a continuous function f(r) on 1-1, 1] that is not identically zero such that f(r)" dr = 0 for every positive odd integer n? (c) Does there exist a continuous function j(r) on [a, b] that is not identically zero such that j(x)" dr = 0 for every positive odd integer n? (Hint: adapt your solution to part (b).) (d) Let k(r) be a continuous function on [a, b]. Suppose that for every continuous function () on [a, b], we have [k(r)f(x) dr = 0. Prove that k(r) is identically zero.