4. UNDERSTANDING WHY WE NEED Vg 0. Your aim is to minimize the function f (x, y) = 3x subject to the constraint x + y? – x³ = 0. (a) Show that if (x, y) satisfies the given constraint, we must have x > 0. Use this fact to find the absolute minimum of f(x, y) = 3x subject to the constraint. (b) Write out the Lagrange multiplier equations for this problem. Show that for any point (x, y) satisfying the Lagrange multiplier equations with A, we must have A # 0 and x + 0. Conclude that y = 0 at such a point. (c) Suppose (x, y) satisfies the Lagrange multiplier equations. From the previous parts, the point is of the form (x, 0) for x > 0. For which x is f(x,0) the absolute minimum value of f along the constraint? Show that the Lagrange multiplier equations do not hold at this point. What is Vg at this point? (d) When two vectors are parallel, one vector is a multiple of the other. It doesn't always go both ways! For example v = (1, 2) is parallel to w = 0, and in fact w = 0 v; but v is not a scalar multiple of w! Summarize: Recall that the Lagrange condition comes from Vƒ parallel to Vg; this can be written Vƒ = \Vg so long as we assume what about Vg?
4. UNDERSTANDING WHY WE NEED Vg 0. Your aim is to minimize the function f (x, y) = 3x subject to the constraint x + y? – x³ = 0. (a) Show that if (x, y) satisfies the given constraint, we must have x > 0. Use this fact to find the absolute minimum of f(x, y) = 3x subject to the constraint. (b) Write out the Lagrange multiplier equations for this problem. Show that for any point (x, y) satisfying the Lagrange multiplier equations with A, we must have A # 0 and x + 0. Conclude that y = 0 at such a point. (c) Suppose (x, y) satisfies the Lagrange multiplier equations. From the previous parts, the point is of the form (x, 0) for x > 0. For which x is f(x,0) the absolute minimum value of f along the constraint? Show that the Lagrange multiplier equations do not hold at this point. What is Vg at this point? (d) When two vectors are parallel, one vector is a multiple of the other. It doesn't always go both ways! For example v = (1, 2) is parallel to w = 0, and in fact w = 0 v; but v is not a scalar multiple of w! Summarize: Recall that the Lagrange condition comes from Vƒ parallel to Vg; this can be written Vƒ = \Vg so long as we assume what about Vg?
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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Transcribed Image Text:4. UNDERSTANDING WHY WE NEED Vg 0. Your aim is to minimize the function f (x, y) = 3x
subject to the constraint x + y? – x³ = 0.
(a) Show that if (x, y) satisfies the given constraint, we must have x > 0. Use this fact to
find the absolute minimum of f(x, y) = 3x subject to the constraint.
(b) Write out the Lagrange multiplier equations for this problem. Show that for any point
(x, y) satisfying the Lagrange multiplier equations with A, we must have A # 0 and
x + 0. Conclude that y = 0 at such a point.
(c) Suppose (x, y) satisfies the Lagrange multiplier equations. From the previous parts, the
point is of the form (x, 0) for x > 0. For which x is f(x,0) the absolute minimum value
of f along the constraint? Show that the Lagrange multiplier equations do not hold at
this point. What is Vg at this point?
(d) When two vectors are parallel, one vector is a multiple of the other. It doesn't always
go both ways! For example v = (1, 2) is parallel to w = 0, and in fact w = 0 v; but v
is not a scalar multiple of w!
Summarize: Recall that the Lagrange condition comes from Vƒ parallel to Vg; this can
be written Vƒ = \Vg so long as we assume what about Vg?
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