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**Unformatted text preview: **( x ) for all x , there is some number N such that f ( x ) > e kx for all x > N . 6. Put 92. A4. Let f be an inﬁnitely differentiable real-valued function deﬁned on the real numbers. If f ± 1 n ² = n 2 n 2 + 1 , n = 0 , 1 , 2 ,... compute the values of the derivatives f ( k ) (0) for all positive integers k . 7. Put 93. B4. The function K ( x,y ) is positive and continuous for ≤ x ≤ 1 , ≤ y ≤ 1 , and the functions f ( x ) and g ( x ) are positive and continuous for ≤ x ≤ 1 . Suppose that Z 1 f ( y ) K ( x,y ) dy = g ( x ) and Z 1 g ( y ) K ( x,y ) dy = f ( x ) for all ≤ x ≤ 1 . Show that f ( x ) = g ( x ) for ≤ x ≤ 1 . 8. Put 96. A6. Let c > be a constant. Give a complete description, with proof, of the set of all continuous functions f : R → R such that f ( x ) = f ( x 2 + c ) for all x ∈ R ....

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- Fall '14
- NORIN
- Math, Calculus, Derivative, Mean Value Theorem, Continuous function, differentiable function