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Unformatted text preview:  x  , for large values of  x  the function sin x(1+ x 2 )1 is positive near the points / 2+2 n and is negative near the points/ 2 + 2 n . By the intermediate value theorem, there is a solution in each interval [/ 2 + 2 n,/ 2 + 2 n ], where n = 1 , 2 ,... 5. continuous and unbounded on [0 , 1); YES e.g. f ( x ) = 1 / (1x ). 6. continuous and unbounded on [0 , 1]; NO, the range of a continuous function on a closed bounded interval is a closed bounded interval. 7. continuous on [0 , 1] and { f ( x ) : x [0 , 1] } = (0 , 1); NO, for the same reason 8. continuous on (0 , 1) and { f ( x ) : x (0 , 1) } = [0 , 1]; YES, e.g. f ( x ) = sin(4 x ). 9. continuous on (0 , 1) and { f ( x ) : x (0 , 1) } = [0 , 1] [3 , 4]. NO, by the intermediate value theorem. 1...
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