Question

Suppose Alexander would like to maximize a continuously differentiable function f : R3 → R on the set S = {(x, y, z) ER3 : x + y + z = 0} : Alexander knows all of the following: • The gradient of f never vanishes inside {(x, y, z) E R3 : x + y + z < 0} • There is exactly one solution x, y, z, 2 e Rto the system of equations Vf(x, y, z) = 1(1, 1, 1); x + y + z = 0 Namely (x, y, z) = (2, 3, -5) and 2 = 7 = = Which of the following must be true?

a. The function has a global maximum on S at (x, y, z) = (2, 3, -5).
b. The function does not have a global maximum on S.
c. The function has a global maximum but it may not be at (x, y, z) = (2, 3, -5).
d. None of the above are true.

Answer 1

b. The function does not have a global maximum on S.

The question provides information about a critical point
`\( (2, 3, -5) \)`where the gradient of
`\( f \) is \( \langle 1, 1, 1 \rangle \).` However, without knowledge of the Hessian matrix and its definiteness at this point, we cannot determine whether it is a global maximum. Therefore, the function may not have a global maximum on
`\( S \) at \( (2, 3, -5) \)`, leading to the selection of option b.

Step-by-step explanation:

In the given scenario, Alexander is maximizing a continuously differentiable function
`\( f : \mathbb{R}^3 \rightarrow \mathbb{R} \)`on the set
`\( S = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\} \).` The gradient of
`\( f \)` never vanishes inside
`\( \{(x, y, z) \in \mathbb{R}^3 : x + y + z < 0\} \),`and there is a unique solution
`\( (x, y, z) = (2, 3, -5) \)`to the system of equations
`\( \\abla f(x, y, z) = \langle 1, 1, 1 \rangle \) and \( x + y + z = 0 \).`

Now, to determine whether there is a global maximum on
`\( S \) at \( (2, 3, -5) \)`, we can examine the Hessian matrix. The Hessian matrix
`\( H \)` is the matrix of second-order partial derivatives o
`f \( f \). If \( H \)` is positive definite at
`\( (2, 3, -5) \), then \( f \)` has a local minimum; if negative definite, it has a local maximum. However, the question does not provide information about the definiteness of the Hessian, so we cannot conclusively state the nature of the critical point.

Therefore, we cannot assert that the function has a global maximum at
`\( (2, 3, -5) \)`or anywhere else on
`\( S \).`Thus, the correct answer is option b: The function does not have a global maximum on
`\( S \).`