WS6s

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MATH1821 Mathematical Methods for Actuarial Science I Worksheet 6 1. Assume that f is continuous on [ a, b ] and differentiable on ( a, b ). Also assume that f ( a ) and f ( b ) have opposite signs and that f ̸ = 0 between a and b . Show that f ( x ) = 0 exactly once between a and b . Solution. The existence of a solution to f ( x ) = 0 is guaranteed by the Intermediate Value Theorem. Now, assume on the contrary that f ( x 1 ) = f ( x 2 ) = 0 for x 1 < x 2 . Then by applying Rolle’s Theorem to f on [ x 1 , x 2 ], we get a c ( x 1 , x 2 ) such that f ( c ) = 0, contradicting the assumption. 2. What values of a and b make f ( x ) = x 3 + ax 2 + bx have (a) a local maximum at x = - 1 and a local minimum at x = 3? (b) a local minimum at x = 4 and a point of inflection at x = 1? Solution. First of all, f ( x ) = 3 x 2 + 2 ax + b and f ′′ ( x ) = 6 x + 2 a . (a) From f ( - 1) = f (3) = 0, we get a = - 3 and b = - 9. We check that
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