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(Solved): 1. (a) Show that g(x)=2x+x1 has a unique fixed point on the interval [1,2]. (b) To solve the ...
![1. (a) Show that \( g(x)=\frac{x}{2}+\frac{1}{x} \) has a unique fixed point on the interval \( [1,2] \).
(b) To solve the eq](https://media.cheggcdn.com/study/f61/f6150579-fbaf-4c67-9eb9-f5d5a44ff478/image)
1. (a) Show that has a unique fixed point on the interval . (b) To solve the equation , derive a fixed-point method using (a), and compute starting at (c) Show that is approximating the root of the given equation in within .
Expert Answer
Answer given below ???? (a) To show that g(x)= x/2 + 1/x has a unique fixed point on the interval [1,2], we need to demonstrate that there exists a value c in [1,2] such that g(c) = c, and that this value is unique. First, note that g(1) = 1/2 + 1/1 = 3/2, and g(2) = 2/2 + 1/2 = 5/2. Since g(x) is continuous on the interval [1,2], by the intermediate value theorem, there exists at least one c in [1,2] such that g(c) = c. Next, we need to show that this fixed point is unique. Suppose there exist two fixed points c1 and c2, where c1 ? c2. Then we have g(c1) = c1 and g(c2) = c2. Without loss of generality, assume c1 < c2. Then we have: g(c1) < g(c2) c1/2 + 1/c1 < c2/2 + 1/c2 c1c2 + 2 < c1c2/2 + c1 + c2/2 2c1c2 + 4 < c1c2 + c1 + c2 c1c2 < 4 But this is a contradiction since c1 and c2 are both in [1,2], so their product cannot be less than 1 or greater than 4. Therefore, we must have a unique fixed point in the interval [1,2].