Consider the function f(x, y) = x²+(y-2)², contours of which are shown in the left panel below. We seek the global maximum and minimum of f on or inside the region depicted by the blue piecewise curve, where C1 = C2 = C3 = D = {(x,y) R²² +3y=9, z € (−√3,3)} {(x, y) R2 | y=0, x = (1,3)} {(x,y) € R² | 2x+(1 + √3) y = 2, x € (−√√3,1)} {(x, y) € R² | x² + 3y < 9, y > 0, 2x + (1+ √3) y > 2}. The three vertices are V₁ = (-√√3,2), V₂ = (1,0), and V3 = (3,0). A graph of the function, restricted to the specified region, is shown in the right panel below. 2 Vi D C₁ V2 C2 11 1 2 11 V3 10 f(x,y) Perform your analysis in the domain using the following steps: (i) Region D: Solve Vf = 0 to find the unconstrained maxima/minima of ƒ and use the second derivative test to determine their nature. Reject any critical points external to D. (ii) Segment C₁: (A) Construct a single variable function g(x) = f(x, y), where y is replaced by the definition for C₁ in terms of x. (B) Find any critical points by solving g'(x) = 0. (C) Determine the nature of the critical points with single-variable second derivative analysis. (iii) Segment C3: Find the total derivative dr df - -√f. then set it to 0 to find any critical points. dt dt C3 has the parametric definition -2 r(t) = ti+ (t-1), 1+√3 (iv) Segment C2: Use the appropriate component of Vf to find any critical points on C2. (This can also be interpreted as a directional derivative or total derivative being set to 0, as in (iii)) (v) Vertices V1, V2, V3: Compute ƒ at the 3 vertices. (vi) Summarise your results from (i)-(v) with the corresponding values of ƒ and give the global maximum and minimum.
Consider the function f(x, y) = x²+(y-2)², contours of which are shown in the left panel below. We seek the global maximum and minimum of f on or inside the region depicted by the blue piecewise curve, where C1 = C2 = C3 = D = {(x,y) R²² +3y=9, z € (−√3,3)} {(x, y) R2 | y=0, x = (1,3)} {(x,y) € R² | 2x+(1 + √3) y = 2, x € (−√√3,1)} {(x, y) € R² | x² + 3y < 9, y > 0, 2x + (1+ √3) y > 2}. The three vertices are V₁ = (-√√3,2), V₂ = (1,0), and V3 = (3,0). A graph of the function, restricted to the specified region, is shown in the right panel below. 2 Vi D C₁ V2 C2 11 1 2 11 V3 10 f(x,y) Perform your analysis in the domain using the following steps: (i) Region D: Solve Vf = 0 to find the unconstrained maxima/minima of ƒ and use the second derivative test to determine their nature. Reject any critical points external to D. (ii) Segment C₁: (A) Construct a single variable function g(x) = f(x, y), where y is replaced by the definition for C₁ in terms of x. (B) Find any critical points by solving g'(x) = 0. (C) Determine the nature of the critical points with single-variable second derivative analysis. (iii) Segment C3: Find the total derivative dr df - -√f. then set it to 0 to find any critical points. dt dt C3 has the parametric definition -2 r(t) = ti+ (t-1), 1+√3 (iv) Segment C2: Use the appropriate component of Vf to find any critical points on C2. (This can also be interpreted as a directional derivative or total derivative being set to 0, as in (iii)) (v) Vertices V1, V2, V3: Compute ƒ at the 3 vertices. (vi) Summarise your results from (i)-(v) with the corresponding values of ƒ and give the global maximum and minimum.
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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--- Please answer all parts of the attached question ---
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Transcribed Image Text:Consider the function f(x, y) = x²+(y-2)², contours of which are shown in the left panel below. We
seek the global maximum and minimum of f on or inside the region depicted by the blue piecewise
curve, where
C1
=
C2 =
C3
=
D =
{(x,y) R²² +3y=9, z € (−√3,3)}
{(x, y) R2 | y=0, x = (1,3)}
{(x,y) € R² | 2x+(1 + √3) y = 2, x € (−√√3,1)}
{(x, y) € R² | x² + 3y < 9, y > 0, 2x + (1+ √3) y > 2}.
The three vertices are V₁ = (-√√3,2), V₂ = (1,0), and V3 = (3,0).
A graph of the function, restricted to the specified region, is shown in the right panel below.
2
Vi
D
C₁
V2
C2
11
1
2
11
V3
10
f(x,y)
Perform your analysis in the domain using the following steps:
(i) Region D: Solve Vf = 0 to find the unconstrained maxima/minima of ƒ and use the second
derivative test to determine their nature. Reject any critical points external to D.
(ii) Segment C₁:
(A) Construct a single variable function g(x) = f(x, y), where y is replaced by the definition for
C₁ in terms of x.
(B) Find any critical points by solving g'(x) = 0.
(C) Determine the nature of the critical points with single-variable second derivative analysis.
(iii) Segment C3: Find the total derivative
dr
df
-
-√f. then set it to 0 to find any critical points.
dt
dt
C3 has the parametric definition
-2
r(t) = ti+
(t-1),
1+√3
(iv) Segment C2: Use the appropriate component of Vf to find any critical points on C2. (This can
also be interpreted as a directional derivative or total derivative being set to 0, as in (iii))
(v) Vertices V1, V2, V3: Compute ƒ at the 3 vertices.
(vi) Summarise your results from (i)-(v) with the corresponding values of ƒ and give the global
maximum and minimum.
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