Question at position 5 The critical point of f(x,y)=x2−4y2+3z2f\left(x,y\right)=x^2-4y^2+3z^2 subject to the constraint x+16y−9z=36x+16y-9z=36 is(1, -3, 4)(3, 1, -4)(-1, 4, 3)(4, -3, 1)(4, 1, -3)Single choice
A
(1, -3, 4)
B
(3, 1, -4)
C
(-1, 4, 3)
D
(4, -3, 1)
E
(4, 1, -3)

Question

Question at position 5 The critical point of f(x,y)=x2−4y2+3z2f\left(x,y\right)=x^2-4y^2+3z^2 subject to the constraint x+16y−9z=36x+16y-9z=36 is(1, -3, 4)(3, 1, -4)(-1, 4, 3)(4, -3, 1)(4, 1, -3)

BlackTom AI · Accepted Answer

We restate the problem in our own words: Find the point (x, y, z) on the plane x + 16y − 9z = 36 where the function f(x, y, z) = x^2 − 4y^2 + 3z^2 has a critical point subject to the constraint.

Step 1: Apply the method of Lagrange multipliers. The gradient of f is ∇f = (2x, −8y, 6z). The gradient of the constraint g(x, y, z) = x + 16y − 9z − 36 is ∇g = (1, 16, −9). At a constrained critical point, ∇f = λ ∇g for some scalar λ. This gives the system:
- 2x = λ(1) → x = λ/2
- −8y = λ(16) → y = −2λ
- 6z = λ(−9) → z = −(3/2)λ

Step 2: Impose the constraint x + 16y − 9z = 36 with these expressions in terms of λ:
x + 16y − 9z = (λ/2) + 16(−2λ) − 9(−3λ/2)
= (λ/2) − 32λ + (27/2)λ
= (28/2)λ − 32λ = 14λ − 32λ = −18λ.
Setting this equal to 36 gives −18λ = 36, so λ = −2.

Step 3: Back-substitute λ into the expressions for x, y, z:
x = λ/2 = −1,
y = −2λ = 4,
z = −(3/2)λ = 3.
Thus the candidate point is (−1, 4, 3).

Step 4: Check each provided option against the Lagrange system and the constraint.
- Option (1, −3, 4): From x = λ/2 we get λ = 2, then y should equal −2λ = −4, which does not match y = −3. Therefore this option is not consistent with the multiplier equations.
- Option (3, 1, −4): From x = λ/2 we get λ = 6, then y should be −2λ = −12, which does not match y = 1. Therefore this option is not consistent.
- Option (−1, 4, 3): From x = λ/2 we get λ = −2, then y should be −2λ = 4 and z should be −(3/2)λ = 3, which exactly matches y and z. Also, check the constraint: x + 16y − 9z = (−1) + 64 − 27 = 36, which satisfies the constraint. Therefore this option is consistent with the Lagrange conditions and the constraint.
- Option (4, −3, 1): From x = λ/2 we get λ = 8, then y should be −2λ = −16, which does not match y = −3.
- Option (4, 1, −3): From x = λ/2 we get λ = 8, then y should be −2λ = −16, which does not match y = 1.

Conclusion from the checks: only the third option satisfies both the Lagrange multiplier equations and the constraint, namely (−1, 4, 3).

Question at position 5 The critical point of f(x,y)=x2−4y2+3z2f\left(x,y\right)=x^2-4y^2+3z^2 subject to the constraint x+16y−9z=36x+16y-9z=36 is(1, -3, 4)(3, 1, -4)(-1, 4, 3)(4, -3, 1)(4, 1, -3)Single choice

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