A proof of Theorem 1, page 268, is outlined below. The

Chapter 5, Problem 30E

(choose chapter or problem)

A proof of Theorem 1 , page 268 , is outlined below. The goal is to show that \(f\left(x^{*}, y^{*}\right)=g\left(x^{*}, y^{*}\right)=0\) Justify each step.

(a) From the given hypotheses, deduce that \(\lim _{t \rightarrow \infty} x^{\prime}(t)=f\left(x^{*}, y^{*}\right)\) and \(\lim _{t \rightarrow \infty} y^{\prime}(t)=(t)=g\left(x^{*}, y^{*}\right)\).

(b) Suppose \(f\left(x^{*}, y^{*}\right)>0\). Then, by continuity, \(x^{\prime}(t)>f\left(x^{*}, y^{*}\right) / 2\)  for all large \(t\) (say, for \(t \geq T\)) Deduce from this that \(x(t)>t f\left(x^{*}, y^{*}\right) / 2+C\) for \(t>T\), where \(C\) is some constant.

(c) Conclude from part (b) that \(\lim _{t \rightarrow \infty} x(t)=+\infty\), contradicting the fact that this limit is the finite number \(x^{*}\). Thus, \(f\left(x^{*}, y^{*}\right)\) cannot be positive.

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

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

Equation Transcription:

 

Text Transcription:

f(x^*,y^*)=g(x^*,y^*)=0

lim_t right arrow infinity x'(t)=f(x^*,y^*)

lim_t right arrow infinity y'(t)=(t)=g(x^*,y^*)

f(x^*,y^*)>0

x'(t)>f(x^*,y^*)/2

t

T geq T

x(t)>tf(x^*,y^*)/2+C

t>T

C

lim_t right arrow infinity x(t)=+infinity

x^*

f(x^*,y^*)

f(x^*,y^*)<0

(x^*,y^*)