Question

Let \( f(x, y, z) = x^3 + y^3 + z^3 - 3xyz. \) A point at which the gradient of \( f \) is equal to zero is

• A. \((-1, 1, -1)\)
• B. \((-1, -1, -1)\)
• C. \((-1, 1, 1)\)
• D. \((1, -1, 1)\)

Hint:

For symmetric functions like \( x^3 + y^3 + z^3 - 3xyz \), gradient vanishes when all variables are equal (i.e., \( x = y = z \)).

Answer 1

The Correct Option is B

Step 1: Compute the gradient.
\[ \nabla f = (3x^2 - 3yz, 3y^2 - 3xz, 3z^2 - 3xy). \]

Step 2: Set each component equal to zero.
\[ 3x^2 - 3yz = 0, 3y^2 - 3xz = 0, 3z^2 - 3xy = 0. \] Simplify: \[ x^2 = yz, y^2 = xz, z^2 = xy. \]

Step 3: Analyze possible solutions.
If \( x = y = z \), all equations hold trivially. So possible points are \( (1, 1, 1) \) and \( (-1, -1, -1). \)

Step 4: Check sign consistency.
Substitute \( (-1, -1, -1) \): \[ x^2 = 1 = yz = (-1)(-1) = 1. \] All equations hold.

Final Answer: \((-1, -1, -1)\).