Answer:
C. Neither
Step-by-step explanation:
To determine whether the function f(x, y) = 10x²y² + 8x⁴y⁴ has a maximum, a minimum, or neither at the origin (x = 0, y = 0), we need to analyze the function's behavior in the vicinity of the origin.
Let's calculate the partial derivatives of f(x, y) with respect to x and y:
∂f/∂x = 20xy² + 32x³y⁴
∂f/∂y = 20yx² + 32x⁴y³
Next, we evaluate these partial derivatives at the origin:
∂f/∂x(0, 0) = 0
∂f/∂y(0, 0) = 0
The partial derivatives are both zero at the origin, which means we cannot determine the nature of the critical point using only the first-order derivatives.
To further investigate, we can calculate the second-order partial derivatives:
∂²f/∂x² = 20y² + 96x²y⁴
∂²f/∂y² = 20x² + 96x⁴y²
∂²f/∂x∂y = 40xy + 128x³y³
Evaluating the second-order partial derivatives at the origin gives:
∂²f/∂x²(0, 0) = 0
∂²f/∂y²(0, 0) = 0
∂²f/∂x∂y(0, 0) = 0
To determine the nature of the critical point at the origin, we need to consider higher-order derivatives or use alternative methods such as the Hessian matrix or Taylor series expansion.
However, since the second-order partial derivatives are all zero at the origin, we cannot conclude whether it is a maximum, minimum, or neither using this information alone.
Therefore, the correct answer is C. Neither.
