do both of them please! Use Lagrange multipliers to find the maximum value of th
ID: 2841710 • Letter: D
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do both of them please!
Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. f(x, y) = 8x2 - 4y2, 8x2 + 4y2 = 9 Select the correct answer. f(x, y) = 1/9 f(x, y) = 1/4 f(x, y) = 1/8 f(x, y) = 8 f(x,y) = 9 Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. f(x, y, z) = 14x + 8y + 12z, x2 + y2 + z2 = 101 Select the correct answer. f(20, 9, 18) = 568 f(14, 8, 12) = 404 f(8, 17, 5) = 308 f(7, 4, 6) = 202 f(8, 4, 12) = 212Explanation / Answer
19) maximize f(x, y) = 8x^2 - 4y^2 subject to g(x, y) = 8x^2 + 4y^2 = 9
<fx, fy> = L<gx, gy>
<16x, -8y> = L<16x + 8y>
system of equations:
16x = 16xL
-8y = 8yL
8x^2 + 4y^2 = 9
16xL - 16x = 0
16x(L - 1) = 0
x = 0, L = 1
8yL + 8y = 0
8y(L + 1) = 0
y = 0, L = -1
If x = 0, then 8x^2 + 4y^2 = 9 gives 4y^2 = 9 or y^2 = 9/4 or y = +/- 3/2
(0, -3/2) and (0, 3/2)
If y = 0, then 8x^2 + 4y^2 = 9 gives 8x^2 = 9 or x^2 = 9/8 or x = +/-3/sqrt(8))
(3/sqrt(8), 0) and (-3/sqrt(8), 0)
Plugging into f(x, y) gives:
f(0, -3/2) = 0 - 4(-3/2)^2 = -4(9/4) = -36
f(0, 3/2) = 0 - 4(3/2)^2 = -36
f(3/sqrt(8), 0) = 8(3/sqrt(8))^2 - 0 = 8(9/8) = 9
f(-3/sqrt(8), 0) = 8(-3/sqrt(8))^2 - 0 = 8(9/8) = 9
max value of f(x, y) = 9
20) maximize f(x, y, z) = 14x + 8y + 12z subject to g(x, y, z) = x^2 + y^2 + z^2 = 101
<fx, fy, fz> = L<gx, gy, gz>
<14, 8, 12> = L<2x, 2y, 2z>
system of equations:
14 = 2xL
8 = 2yL
12 = 2zL
x^2 + y^2 + z^2 = 101
solve for x, y, z in terms of L:
7/L = x
4/L = y
6/L = z
(7/L)^2 + (4/L)^2 + (6/L)^2 = 101
49/L^2 + 16/L^2 + 36/L^2 = 101
49 + 16 + 36 = 101L^2
101 = 101L^2
L = -1, 1
L = -1 gives (-7, -4, -6)
L = 1 gives (7, 4, 6)
plugging in f(x, y, z):
f(-7, -4, -6) = 14(-7) + 8(-4) + 12(-6) = -202
f(7, 4, 6) = 14(7) + 8(4) + 12(6) = 202
max value of f(x, y, z) = 202
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