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a4 24

# a4 24 - MATH 242 Analysis 1(Fall 2011 Assignment 4 Due in...

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MATH 242 Analysis 1 (Fall, 2011) Assignment 4. Due in class Tuesday, November 8. Instructions: This assignment is out of 50 points. You may choose any questions whose points add up to 50. As in the previous assignment, provide all details in your work. Also as previously, stars (*) indicate possibly more challenging questions. 1. (10 pts.) (a) Let f ( x ) = ( x 2)( x 3)( x 4)( x 5)( x 6). If the Bisection Method is applied, beginning with the interval [0 , 7], which of the roots of f is located ? (b) Show that the equation 1 x 2 + x + x 2 = 2 x has at least two solutions x > 0. 2. (10 pts.) Let f : [0 , 1] R be continuous and such that f (0) = f (1). Prove that there exists a point c [0 , 1 2 ] such that f ( c ) = f ( c + 1 2 ). [Hint: Consider g ( x ) = f ( x ) f ( x + 1 2 ).] Conclude that there are, at any time, antipodal points on the earth’s equator that have the same temperature. 3. (10 pts.) Let a < b and f : ( a,b ) R be continuous and such that lim x a f ( x ) = 0 = lim x b f ( x ). Prove that f is bounded on ( a,b ) and attains either a maximum or a minimum on ( a,b ).

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a4 24 - MATH 242 Analysis 1(Fall 2011 Assignment 4 Due in...

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