Question

For the function \(f(x, y) = x^3 + x^2 y^2\), the values of \(f_x(-1, -1)\) and \(f_y(-1, -1)\) are respectively:

• A. 2, 1
• B. -1, 2
• C. 1, -2
• D. 2, -1

Hint:

Exam Tip:
For partial derivatives:

• Treat all other variables as constants.
• Use the power rule for differentiation.
• Evaluate at the given point.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find the partial derivatives of \(f(x, y) = x^3 + x^2 y^2\) and evaluate them at \((-1, -1)\).

Step 2: Key Formula or Approach:
\[ f_x = \frac{\partial f}{\partial x} = 3x^2 + 2x y^2 \] \[ f_y = \frac{\partial f}{\partial y} = 2x^2 y \]

Step 3: Detailed Explanation:
Evaluate at \((-1, -1)\): \[ f_x(-1, -1) = 3(-1)^2 + 2(-1)(-1)^2 = 3 - 2 = 1 \] \[ f_y(-1, -1) = 2(-1)^2(-1) = -2 \] So, \(f_x = 1\) and \(f_y = -2\).
This matches option (C).

Step 4: Final Answer:
Therefore, option (C) is correct.