Exercise 2.1. Let f(x) be a twice-continuously differentiable univariate function defined on a domain (a, b). Assume that f has a zero a* € (a, b) and that f"(x) > 0 and f'(x) < 0 for all æ € (a, b). Consider the application of Newton's method to find a zero of f, with ro chosen so that r, € (a, b) with f(x1) > 0. (a) Show that xị < x*. (b) Show that if æk € (a, b) with f(®k) > 0, then æk < *k+1 < x*. (c) Show that if ær E (a, b) and f(x4) > 0, then f(xk+1) > 0.

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.