Question

The number of critical points of the function \[ f(x, y) = (x^2 + 3y^2)^2 e^{-(x^2 + y^2)} \] is .....................

Hint:

To find critical points, compute the partial derivatives, set them equal to zero, and solve for the values of \( x \) and \( y \).

Answer 1

Correct Answer: 5

Step 1: Find the partial derivatives.
The critical points of the function occur where the gradient \( \nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \) is zero. First, compute the partial derivatives. - The partial derivative with respect to \( x \) is: \[ \frac{\partial f}{\partial x} = 2(x^2 + 3y^2) \cdot 2x e^{-(x^2 + y^2)} - 2x(x^2 + 3y^2)^2 e^{-(x^2 + y^2)}. \] - The partial derivative with respect to \( y \) is: \[ \frac{\partial f}{\partial y} = 2(x^2 + 3y^2) \cdot 6y e^{-(x^2 + y^2)} - 2y(x^2 + 3y^2)^2 e^{-(x^2 + y^2)}. \]
Step 2: Set the partial derivatives equal to zero.
Set both partial derivatives to zero to find the critical points. From the equations, we solve for \( x \) and \( y \). This leads to solutions at \( (0,0) \).
Step 3: Check for the number of critical points.
Since both derivatives are zero at the origin, we conclude there is one critical point.
Step 4: Conclusion.
The number of critical points is: \[ \boxed{1}. \]