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Unformatted text preview: does, then it must pass through zero). 4. 4.15.2 (p186) Let f ( x ) = x 3-3 x + b . Suppose x 1 , x 2 are zeros of f ( x ) in the interval-1 x 1. Then by the Rolles theorem, there exists a point c in ( x 1 , x 2 ) such that f ( c ) = 0. But f ( x ) = 3 x 2-3, so f = 0 only when x = 1, contradiction. 5. 4.21.4 (p194) Fix y , let f ( x ) = x 2 + y 2 where x, y > 0. Observe that f ( x ) = x 2 + ( S-x ) 2 , hence f ( x ) = 2 x-2( S-x ). Hence f ( x ) = 0 only when x = S-x = y . Finally, note that f 00 ( x ) = 4 > 0, hence f attains its minimum when x = y ....
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