1.) Find the maximum and minimum values of F (x, y) = 2x + y on the ellipse x^2
ID: 2837054 • Letter: 1
Question
1.) Find the maximum and minimum values of F (x, y) = 2x + y on the ellipse x^2 + 16y^2 = 1
A.) Maximum value: ...............................................?
B.) Minimum value: ................................................?
2.) Find the maximum and minimum values of F (x, y, z) = 1x + 3y + 4z on the sphere
x^2 + y^2 + z^2 = 1
A.) Maximum value:...................................................?
B.) Minimum value:......................................................?
3.) Find the Maximum and Minimum values of F (x, y) = 2x + y on the ellipse x^2 + 9y^2 = 1
A.) Maximum value: ......................................................?
B.) Minimum value: ..........................................................?
4.) Find the maximum and minimum values of F (x, y) = 13x^2 + 14y^2 on the disk
D: x^2 + y^2 (Less than or equal to) 1
A.) Maximum value: .....................................................?
B.) Minimum value: .....................................................?
5.) Find the maximum and minimum values of the function F (x, y, z) = x + 2y subject to the constraints y^2 + z^2 = 100 and x + y + z = 6.
A.) Maximum value is:...............................................? occuring at (....................... , ........................ , ...................)
B.) Minimum values is: .................................................? occuring at (................ , .......................... , .....................)
Explanation / Answer
1.) Find the maximum and minimum values of F (x, y) = 2x + y on the ellipse x^2 + 16y^2 = 1
F = 2x+y , G = x^2 +16y^2 -1 = 0
use Lagrange's multipliers method
Fx = 2, Fy = 1
Gx = 2x , Gy = 32y
Fx/Gx = Fy/Gy
2/2x = 1/32y
32y = x
y = x/32
substitute y=x/32 in G
x^2 + 16(x/32)^2 = 1
x^2 + 16*x^2/32*32 = 1
x^2 + x^2/64 =1
65x^2 =64
x^2 = 64/65
x = 8/sqrt(65), -8/sqrt(65)
y = x/32 = 8/{32*sqrt(65)} = 1/4sqrt(65) ,
y= -8/32sqrt(65) = -1/4sqrt(65)
critical points (8/sqrt(65) , 1/4sqrt(65)),-(8/sqrt(65) , -1/4sqrt(65))
maximum is at (8/sqrt(65) , 1/4sqrt(65))
F(8/sqrt(65) , 1/4sqrt(65)) = 2*8/sqrt(65) + 1/4sqrt(65)
= 65/4sqrt(65)
maximum value = sqrt(65) /4
minimum is at(-8/sqrt(65) , -1/4sqrt(65))
F(-8/sqrt(65) , -1/4sqrt(65)) = 2*-8/sqrt(65) + (-1)/4sqrt(65)
= -65/4sqrt(65)
Minimum value is = -sqrt(65)/4
2.) Find the maximum and minimum values of F (x, y, z) = 1x + 3y + 4z on the sphere
x^2 + y^2 + z^2 = 1
F = x+3y+4z , P = x^2 +y^2 +z^2 -1 = 0
Fx = 1, Fy = 3,Fz = 4
Px = 2x, Py = 2y, Pz = 2z
use Lagrange's multipliers method
Fx/Px = Fy/Py = Fz/Pz
1/2x = 3/2y = 4/2z
1/2x = 3/2y , 1/2x = 4/2z
y = 3x, z = 4x
substitute these values in P
x^2 +(3x)^2 +(4x)^2 = 1
x^2 +9x^2+16x^2 = 1
x = 1/sqrt(26) , -1/sqrt(26)
y = 3x = 3/sqrt(26, -3/sqrt(26)
z = 4x = 4/sqrt(26), -4/sqrt(26)
critical points are (1/sqrt(26) , 3/sqrt(26) , 4/sqrt(26)) , (-1/sqrt(26) , -3/sqrt(26) , -4/sqrt(26))
Maximum is at (1/sqrt(26) , 3/sqrt(26) , 4/sqrt(26))
maximum value is F(1/sqrt(26) , 3/sqrt(26) , 4/sqrt(26)) = 1/sqrt(26) + 9/sqrt(26) + 16/sqrt(26)
= 26/sqrt(26) = sqrt(26)
Minimum is at (-1/sqrt(26) , -3/sqrt(26) , -4/sqrt(26))
Minimum value is F(-1/sqrt(26) , -3/sqrt(26) , -4/sqrt(26)) = -1/sqrt(26) - 9/sqrt(26) - 16/sqrt(26)
= -sqrt(26)
3.) Find the Maximum and Minimum values of F (x, y) = 2x + y on the ellipse x^2 + 9y^2 = 1
F = 2x+y , Q = x^2 +9y^2 -1 =0
Fx = 2, Fy = 1,
Qx = 2x, Qy = 18y
Lagrange's multipliers method
Fx/Qx = Fy/Qy
2/2x = 1/18y
x = 18y
y = x/18
substitute in Q
x^2 +9x^2 /(18*18) = 1
37x^2 = 36
x = 6/sqrt(37), -6/sqrt(37)
y = x/18 = 1/3sqrt(37) , -1/3sqrt(37)
Critical points = (6/sqrt(37) , 1/3sqrt(37)), (-6/sqrt(37) , -1/3sqrt(37))
Maximum value F(6/sqrt(37 , 3/sqrt(37)) = 2*6/sqrt(37) +1/3sqrt(37)
= 37/3sqrt(37) = sqrt(37) /3
Minimum value is F(-6/sqrt(37 , -1/3sqrt(37)) = -2*6/sqrt(37) -1/3sqrt(37)
= -37/3sqrt(37) = -sqrt(37) /3
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