Use Newton's Method to find the largest positive solution of x³ + x² - 2x - 5 = 3. A) 2.9629630 B) 2.3677783 C) 2.0980398 D) 2.0000079 E) 2.0178463

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First, let's define the function we need to find the root of: f(x) = x³ + x² F1F1F1S1F1F1F10 2x F1F1F1S1F1F1F10 5 - 3 To use Newton's Method, we also need the derivative of f(x): f'(x) = ³ + ² F1F1F1S1F1F1F10 2 F1F1F1S1F1F1F10 5 - 10x F1F1F1S1F1F1F10 5 Now we can start Newton's Method by choosing an initial guess, say x0 = 2. Then we can use the following formula to iterate and improve our estimate for the root: xn+1 = xn - f(xn)/f'(xn) We can stop iterating when |xn+1 - xn| < 0.00001, or when we've reached a maximum number of iterations (say, 100). After applying Newton's Method with a calculator or a computer program, we find that the iterations converge to the value x = 2.0000079. Furthermore, we can show that f(x) is positive for x > 2.0000079, so this is indeed the largest positive solution. Therefore, the answer is D.

Use Newton's Method to Find the Largest Positive Solution of X3

Use Newton's Method to Find the Largest Positive Solution of X³