# Mial terms are also continuous so the sum f is

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mial terms are also continuous, so the sum f is continuous. By the Weierstrass Theorem, this continuous function must have a maximum on the compact pro- duction possibilities set. 5. Let f ( x, y ) = x 2 / ( x 2 + y 2 ) so that f : R 2 \ { (0 , 0) } R . a ) What is the range of f ? Answer: Clearly 0 f ( x, y ) 1. Both endpoints can occur: f (0 , 1) = 0 and f (1 , 0) = 1. The range is [0 , 1]. b ) Is f onto? Answer: No, ran f 6 = R 2 . c ) Is f one-to-one? Answer: No, f (0 , 1) = f (0 , 1 / 2) = 0. d ) Is f continuous? Answer: Yes. It is the quotient of continuous functions and the denominator is non-zero. e ) Compute df Answer: df = 2 xy 2 ( x 2 + y 2 ) 2 , - 2 yx 2 ( x 2 + y 2 ) 2 .
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