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# exercises_11 - Analysis 1 Exercises 11 1 Let f be...

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Analysis 1: Exercises 11 1. Let f be continuous on [ a, b ]. Suppose that f ( x ) = 0 for all rational x [ a, b ]. Prove that f ( x ) = 0 for all x [ a, b ]. 2. Prove that the following equations have solutions in [ - 2 , 0]. (a) x 3 - x + 3 = 0; (b) x 5 + x + 1 = 0. 3. Let f and g be continuous on [ a, b ]. Suppose that f ( a ) < g ( a ) and f ( b ) > g ( b ). Prove that there exists x ( a, b ) such that f ( x ) = g ( x ). 4. Prove that the equation x 4 - 3 x - 9 = 0 has at least two solutions in [ - 2 , 2]. 5. Let f : [0 , 1] [0 , 1] be continuous. Prove that ( x [0 , 1]) ( f ( x ) = x ) . 6. Let I stand for the closed interval I := [ a, b ]. Let f : I R and g : I R be continuous functions on I . Define E := { x I : f ( x ) = g ( x ) } . Suppose that ( x n ) E has the property that x n α for some α R . Show that (a) α I ; (b) α E . 7. Let p ( x ) := a 0 x 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 be a cubic polynomial with real coefficients a 0 , a 1 , a 2 , a 3 R . Let us assume that a 3 > 0. Show that (i) lim x + p ( x ) = + ; (ii) lim x →-∞ p ( x ) = -∞ ; (iii) p has at least one real root.

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