Taylor Series/Newton Raphson Method question

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Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.
 
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I'm assuming you forgot to post a link or quote something.

But your questions can be answered without whatever it is that you forgot. ^ denotes exponentiation, ie in x^y, x is the base and y is the exponent. Argmin(f(x)) is the x at which f(x) is minimized. For examples, 2^3=8 and argmin(x^2)=0.