I just want to fill the blanks 1- (B) If there is a local minimum, what is the v
ID: 3343662 • Letter: I
Question
I just want to fill the blanks
1-
(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N.
(C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N.
(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N.
First point (, ) of type
Second point (, ) of type
Third point (, ) of type
Fourth point (, ) of type
(A) How many critical points does have in ?
(Note, is the set of all pairs of real numbers, or the -plane.)
(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N.
(C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N.
(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N.
(E) What is the maximum value of on ? If there is none, type N.
(F) What is the minimum value of on ? If there is none, type N.
Explanation / Answer
1.
By discriminant, they are referring to the discriminant of the Hessian, or second derivatives
fx = 2x-10
fy = 2y - 2
fxx = 2
fyy = 2
fxy = fyx = 0
The determinant is
2 0
0 2
Thus, the discriminant is 4.
The eigenvalues are both 2, so this is a positive definite matrix and we have a minimum (not asked for)
The discriminant is 4 everywhere, including at the minimum
The derivative equals 0 at 2x - 10 = 0 and 2y - 2 = 0, or (5, 1), so this is the minimum.
x^2 + y^2 - 10x - 2y + 4 =
5^2 + 1^2 - 10*5 - 2*1 + 4 =
25 + 1 - 50 - 2 + 4 = -22
2. f(x,y) = xy(1-7x-2y)=xy -7x^2y -2xy^2
fx = y - 14xy -2y^2 = y(1 - 14x - 2y)
fy = x - 7x^2 -4xy = x(1 - 7x - 4y)
fxx = -14y
fyy = -4x
fxy = 1 - 14x -4y
The Hessian is
-14y 1 - 14x -4y
1 - 14x -4y -4x
fx=y(1 - 14x - 2y)
fy=x(1 - 7x - 4y)
We have (0, 0) as a solution.
As the Hessian is
0 1
1 0
the discriminant is -1, so we have a saddle point.
If x = 0 and y is not equal to 0, then (1-14(0) -2y) = 0
1 = 2y
y = 1/2
The discriminant is
-14(1/2) 1 - 14(0) -4(1/2)
1 - 14(0) -4(1/2) 0
or
-7 -1
-1 0
The discriminant is -1, so this is a saddle point.
If y = 0 and x is not equal to 0, then
1 - 7x -4(0) = 0
1 = 7x
x = 1/7
The Hessian is
-14(0) 1 - 14(1/7) -4(0)
1 - 14(1/7) -4(0) -4(1/7)
or
0 -1
-1 -4/7
The discriminant is -1, so this is a saddle point.
If neither equals 0
1 - 14x - 2y = 0
1 - 7x - 4y = 0
14x + 2y = 1
7x + 4y = 1
Double the bottom equation
14x + 8y = 2
Subtract the top equation from the bottom equation.
6y = 1
y = 1/6
7x + 4y = 1
7x = 1 - 4(1/6)
7x = 1/3
x = 1/21
The Hessian is
-14y 1 - 14x -4y
1 - 14x -4y -4x
-14(1/6) 1 - 14(1/21) - 4(1/6)
1 - 14(1/21) - 4(1/6) -4(1/21)
or
-7/3 -1/3
-1/3 -4/21
The discriminant is (-7/3)(-4/21)-(-1/3)^2 = 28/63 - 1/9 = 4/9 - 1/9 = 1/3
The two diagonal elements are negative.
Thus, our matrix is negative definite and this is a maximum.
0
0
S
0
1/2
S
1/21
1/6
MA
1/7
0
S
3.x^7y - 7x^6 + y
fx = 7x^6y - 42x^5
fy = x^7 + 1
x^7 + 1 = 0, so x^7 = -1, or x = -1
Then, 7x^6y - 42x^5 = 0, so
7y + 42 = 0
7y = -42
y = -6
fxx = 42x^5y - 210x^4
fyy = 0
fxy = 7x^6
The Hessian is
42x^5y - 210x^4 7x^6
7x^6 0
or
42 7
7 0
The discriminant is -49, so this is a saddle point
-1
-6
S
Everything else is left blank
A linear function contained within a polygon has its maxima and minima at the vertices.
20 - 4x + 9y at
(0, 0), (9, 0), (9, 13)
f(0,0) = 20 - 4(0)+ 9(0) = 20
f(9,0) = 20 - 4(9)+ 9(0) = -16
f(9, 13) = 20 - 4(9)+ 9(13) = 101
Min is -16 at (9,0) and max is 101 at (9, 13)
-16
9
0
101
9
13
5.
Use LaGrange multipliers
l(x,y) = 6x + y + L(x^2+9y^2 - 1)
Gradient is
(6+2Lx, 1 + 18Ly)
2Lx = -6
18Ly = -1
Dividing,
x/9y = 6
x = 54y
Substitute into
x^2 + 9y^2 = 1
(54y)^2 + 9y^2 = 1
(18*3y)^2 + (3y)^2 = 1
325*9y^2 = 1
y^2 = 1/(9*325)
y = plus or minus 1/15 sqrt(13)
x = 54 y = 54 * plus or minus 1/15 sqrt(13) = plus or minus 18/5 sqrt(13)
Then, 6x + y = plus or minus 6 * 18/5 sqrt(13) + 1/15 sqrt(13) =
plus or minus 65/3 sqrt(13)
Maximum value
65/(3 sqrt(13)) = 6.00925212577332
Minimum value
-65/(3 sqrt(13)) = -6.00925212577332