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ax + by + c > 0,

ax + by + c > 0,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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You are now allowed to assume that the half-planes determined by the line with the equation ax+by +
c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Using
this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line
and that (s, t) and (u, v) are on the same side of the given line. ‹en construct the parametrized line
through (q, r) and (u, v). Consider the mapping
λ γ
7
→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every
λ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.)

You are now allowed to assume that the half-planes determined by the line with the equation ax+by+ c = 0 correspond to the points (x, y) so that ax+by+c<0 and ax+by+c>0, respectively. Using this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line and that (s, t) and (u, v) are on the same side of the given line. Then construct the parametrized line through (q, r) and (u, v). Consider the mapping Xa(q-qλ + uλ) + b(r = rλ + vλ) + c
Transcribed Image Text:You are now allowed to assume that the half-planes determined by the line with the equation ax+by+ c = 0 correspond to the points (x, y) so that ax+by+c<0 and ax+by+c>0, respectively. Using this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line and that (s, t) and (u, v) are on the same side of the given line. Then construct the parametrized line through (q, r) and (u, v). Consider the mapping Xa(q-qλ + uλ) + b(r = rλ + vλ) + c
and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every A, y(x) > 0 or y(x) < 0, depending on which half-plane the points are on.)
Transcribed Image Text:and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every A, y(x) > 0 or y(x) < 0, depending on which half-plane the points are on.)
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