Problem 30E

A proof of Theorem 1, page 268, is outlined below. The goal is to show that Justify each step.

(a)From the given hypotheses, deduce that and (b) Suppose Then, by continuity, x’(t) > f(x*,y*)/2 for all large t (say, for t T). Deduce from this that for t > T, where C is some constant.

(c) Conclude from part (b) that contradicting the fact that this limit is the finite number x*. Thus,f(x*,y*) cannot be positive.

(d) Argue similarly that the supposition that f(x*,y*) < 0 also leads to a contradiction; hence, f(x*,y*) must be zero.

(e) In the same manner, argue that (g*,y*) must be zero. Therefore,f(x*,y*) = g(x*,y*) = 0 and (x*,y*) is a critical point

Introduction to Chapter 1: Difference of Squares: a -b =(a-b)(a+b) Zero Product Property: ab=0; a, b=0 1.1 Linear Equations are first-degree equations in one variable that contain the equality symbol, “=”. Linear equation in the variable x can be expresses in the standard form, ax+b=c, where a ≠, b ∊ ℝ The domain is the largest set of acceptable input values. Of note, the denominator of a fraction cannot be equal to zero. 1.1.1 Solve ax+b=0, solve for x ax+(b-b)=(0-b) 1. Isolate x by subtracting b from both sides. x/a=-b/a 2. Divide both sides by a x=-b/a 1.1.2 Solve 5x-3=2x+8 -2x -2x 1. Subtract 2x from each side 3x-3=8 +3 +3 2. Ad