Unformatted text preview: = x f ( x ) for all x > 0. [Putnam Exam, 2005, B-3] Exercise 4. Let f be a real function on the real line with a continuous third derivative. Prove that there exists a point a such that f ( a ) · f ( a ) · f 00 ( a ) · f 000 ( a ) ≥ 0. [Putnam Exam, 1998, A-3] Exercise 5. Let p ( x ) be a nonzero polynomial of degree less than 1992 having no noncon-stant factor in common with x 3-x . Let d 1992 dx 1992 ³ p ( x ) x 3-x ´ = f ( x ) g ( x ) for polynomials f ( x ) and g ( x ). Find the smallest possible degree of f ( x ). [Putnam Exam, 1992, B-4]...
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- Fall '10
- Equations, Continuous function, Complex number, Putnam Exam