Can you help with question 2?Notes (1)-(3) for question 2c are attached for reference. SECTION 3.3/ EQUALITY CONSTRAINTS: THE LAGRANGE PROBLEM115ROBMS FOR SECTION 3 2SM 1. The functionf(x1, x2, x3) = x} + x3 + 3x3Extx + Ex1x7+ 7x]x –defined on R has ohly one stationary point. Show that it is a local minimum point.2. (a) Let f be defined for all (x, y) by ƒ (x, y) = x+y³-3xy. Show that (0, 0) and (1, 1) are theonly stationary points, and compute the quadratic form in (9) for ƒ at the stationary points.(b) Check the definiteness of the quadratic form at the stationary points.(c) Classify the stationary points by using (1)(3).SM 3. Classify the stationary points of(a) f(x, y, z) = x² +x?y + y²z+z? – 4z(b) f(x1, x2, x3, X4) = 20x2 + 48x3 + 6x4 + 8x,.x2 – 4x} – 12x3 – x; – 4x3HARDER PROBLEMS4. Suppose f(x, y) has only one stationary point (x*, y*) which is a local minimum point. Is(x*, y*) necessarily a global minimum point? It may be surprising that the answer is no. Provethis by examining the function defined for all (x, y) by f (x, y) = (1 +y)³x²+y². (Hint: Lookat f(x, –2) as x → .)3.3 Equality Constraints: The Lagrange ProblemA general optimization problem with equality constraints is of the formmax (min) f (x1, ..., Xn) subject to1q = ("x'· · 'Ix)18(u > w)(1)"q = ("x' · 'x) 48where the b; are constants. We assume that m < n because otherwise there are usually nodegrees of freedom.In vector formulation, the problem is12:54/18 oy%20Knut%20Sydsaeter,Peter%20Hammond,Atle%20Seierstad, Arne%20Strom%20(z-lib.org).pdfWe next study conditions that allow the stationary points of a function of n variables to beclassified as local maximum points, local minimum points, and saddle points. First recall thesecond-order conditions for local extreme points for functions of two variables. If f (x, y)is a C2 function with (x*, y*) as an interior stationary point, then12)| SH(**, y*) f**, y*)|| *, y°) f*, y" |fi(*, y") > 0 &>0 = local min. at (x*, y*) (1)| K**, y") f(**, y") ||f(**, y) f*, y*) || S(**, y") fi(**, y*) ||**, y*) f**, y") |>0 = local max. at (x*, y*) (2)<0 = (x*, y*) is a saddle point(3)In order to generalize these results to functions of n variables, we need to consider the nleading principal minors of the Hessian matrix f"(x) = (f#(x))nxn:(x)f (x) f(x) (x) S(x) f(x)k = 1,..., n= (x)a(t)(x) STHEOREM 3 2.1 (LOCAL SECOND-ORDER CONDITIONS).(x)l (x)Suppose that f (x) = ƒ(x1, .., Xn) is defined on a set S in R" and that x* is aninterior stationary point. Assume also that f is C² in an open ball around x*. LetDr(x) be defined by (4). Then:(a) Dr(x*) > 0, k = 1,...,nx* is a local minimum point.(b) (-1)* DE(x*) > 0, k = 1,..., nx* is a local maximum point.(c) Dn(x*) #0 and neither (a) nor (b) is satisfiedx* is a saddle point.Show12:51 PM4/18/202

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Description: Can you help with question 2?Notes (1)-(3) for question 2c are attached for reference. SECTION 3.3/ EQUALITY CONSTRAINTS: THE LAGRANGE PROBLEM115ROBMS FOR SECTION 3 2SM 1. The functionf(x1, x2, x3) = x} + x3 + 3x3Extx + Ex1x7+ 7x]x –defined on R has

Here in the given question, the function is given and show that stationary point and compute the…

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2. (a) Let f be defined for all (x, y) by fƒ (x, y) = x'+y' -3xy. Show that (0, 0) and (1, 1) are the only stationary points, and compute the quadratic form in (9) for f at the stationary points. (b) Check the definiteness of the quadratic form at the stationary points.

2. (a) Let f be defined for all (x, y) by fƒ (x, y) = x'+y' -3xy. Show that (0, 0) and (1, 1) are the only stationary points, and compute the quadratic form in (9) for f at the stationary points. (b) Check the definiteness of the quadratic form at the stationary points.

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Can you help with question 2?
Notes (1)-(3) for question 2c are attached for reference.

SECTION 3.3/ EQUALITY CONSTRAINTS: THE LAGRANGE PROBLEM
115
ROB
MS FOR SECTION 3 2
SM 1. The function
f(x1, x2, x3) = x} + x3 + 3x3
Extx + Ex1x7+ 7x]x –
defined on R has ohly one stationary point. Show that it is a local minimum point.
2. (a) Let f be defined for all (x, y) by ƒ (x, y) = x+y³-3xy. Show that (0, 0) and (1, 1) are the
only stationary points, and compute the quadratic form in (9) for ƒ at the stationary points.
(b) Check the definiteness of the quadratic form at the stationary points.
(c) Classify the stationary points by using (1)(3).
SM 3. Classify the stationary points of
(a) f(x, y, z) = x² +x?y + y²z+z? – 4z
(b) f(x1, x2, x3, X4) = 20x2 + 48x3 + 6x4 + 8x,.x2 – 4x} – 12x3 – x; – 4x3
HARDER PROBLEMS
4. Suppose f(x, y) has only one stationary point (x*, y*) which is a local minimum point. Is
(x*, y*) necessarily a global minimum point? It may be surprising that the answer is no. Prove
this by examining the function defined for all (x, y) by f (x, y) = (1 +y)³x²+y². (Hint: Look
at f(x, –2) as x → .)
3.3 Equality Constraints: The Lagrange Problem
A general optimization problem with equality constraints is of the form
max (min) f (x1, ..., Xn) subject to
1q = ("x'· · 'Ix)18
(u > w)
(1)
"q = ("x' · 'x) 48
where the b; are constants. We assume that m < n because otherwise there are usually no
degrees of freedom.
In vector formulation, the problem is
12:5
4/18

oy%20Knut%20Sydsaeter,Peter%20Hammond,Atle%20Seierstad, Arne%20Strom%20(z-lib.org).pdf
We next study conditions that allow the stationary points of a function of n variables to be
classified as local maximum points, local minimum points, and saddle points. First recall the
second-order conditions for local extreme points for functions of two variables. If f (x, y)
is a C2 function with (x*, y*) as an interior stationary point, then
12)
| SH(**, y*) f**, y*)|
| *, y°) f*, y" |
fi(*, y") > 0 &
>0 = local min. at (x*, y*) (1)
| K**, y") f(**, y") |
|f(**, y) f*, y*) |
| S(**, y") fi(**, y*) |
|**, y*) f**, y") |
>0 = local max. at (x*, y*) (2)
<0 = (x*, y*) is a saddle point
(3)
In order to generalize these results to functions of n variables, we need to consider the n
leading principal minors of the Hessian matrix f"(x) = (f#(x))nxn:
(x)f (x) f
(x) (x) S
(x) f
(x)
k = 1,..., n
= (x)a
(t)
(x) S
THEOREM 3 2.1 (LOCAL SECOND-ORDER CONDITIONS).
(x)l (x)
Suppose that f (x) = ƒ(x1, .., Xn) is defined on a set S in R" and that x* is an
interior stationary point. Assume also that f is C² in an open ball around x*. Let
Dr(x) be defined by (4). Then:
(a) Dr(x*) > 0, k = 1,...,n
x* is a local minimum point.
(b) (-1)* DE(x*) > 0, k = 1,..., n
x* is a local maximum point.
(c) Dn(x*) #0 and neither (a) nor (b) is satisfied
x* is a saddle point.
Show
12:51 PM
4/18/202

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