Suppose that f is continuous on [a, b] and fg = 0 for all con- tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there is an obvious g to choose.) Suppose f is continuous on [a, b] and that fg = 0 for those con- tinuous functions g on [a, b] which satisfy the extra conditions g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is an important lemma in the calculus of variations; see the Suggested Reading for references.) Hint: Derive a contradiction from the assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on the behavior of f near xo.
Suppose that f is continuous on [a, b] and fg = 0 for all con- tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there is an obvious g to choose.) Suppose f is continuous on [a, b] and that fg = 0 for those con- tinuous functions g on [a, b] which satisfy the extra conditions g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is an important lemma in the calculus of variations; see the Suggested Reading for references.) Hint: Derive a contradiction from the assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on the behavior of f near xo.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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