Newton’s method Use Newton’s method to find all local extreme ...

Question

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

Accepted Answer

Determine the values of X for all local extreme values of F of X equals X HX using Newton's method. Use the initial value X0 equals 1. Now, to solve this, we first need to find some derivatives. We'll find F of X first, which is given by the product rule. You have X sign H of X. We have our product rule, which gives us Sine H X plus X cosine H X. We can now repeat for our second derivative. Our second derivative, we end up getting the derivative of this. Which is to Plus HX plus X H X. Now that we have our two derivatives, we can make use of our iteration. This tells us X of N + 1 is equals to X of N minus F of X of N. Divided by F of XN. So we can first find our first iteration. X1 Which in our case, will be X0. Minus F of X0 divided by F of X0. We know X0 is 1 as given by the problem. So we have 1 minus. So H Of one? Plus 1 cosine H of 1. Divided by 2, cosine H of 1 plus 1. Sign H of one. You can now approximate this in a calculator to get a value of 0.3621. Now we will repeat with our next iteration. X2 X2 will be given by X1 minus F X1. Divided by F double X1. And we can repeat the process. We have 0.3621. Minus H 0.3621. Plus 0.3621 multiplied by cosine H. Of 0.3621. Divided by 2, cosine H 0.3621. Plus 0.3621, and so on. This simplifies to give us a value of 0.0285. Now from here, we'll just repeat this a few more times, using the same process. If you repeat this, X3 will be approximately 1.5422. Times 10 to -5. X4 Will be approximately 2.4453. Multiply By 10 to 15. X 5 will be a very small number, which is close to 0, and X6 will also be close to 0. Now we can classify our critical point. F double prime. Of 0 Equals 2 cosine H of 0, plus 0 multiplied by sin H of 0. Which gives us 2 This is greater than 0. Because this is greater than 0, F of X equals X. Sign H of X. Has a local minimum. At X equals 0. This is the final answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

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