Question

Let f(x, y) = \(e^{x^2}+y^2\) for (x, y) ∈ \(\R^2\) , and a_n be the determinant of the matrix
\(\begin{pmatrix} \frac{∂^2f}{∂x^2} &  \frac{∂^2f}{∂x∂y} \\  \frac{∂^2f}{∂y∂x} &  \frac{∂^2f}{∂y^2} \end{pmatrix}\)
evaluated at the point (cos(n),sin(n)). Then the limit \(\lim\limits_{n \rightarrow \infin}\) a_n is

• A. non-existent
• B. 0
• C. 6e²
• D. 12e²

Answer 1

The Correct Option is D

To find the limit \(\lim\limits_{n \rightarrow \infty} a_n\), we first need to compute the second-order partial derivatives of the function \(f(x, y) = e^{x^2} + y^2\), and find the determinant of the Hessian matrix at the point \((\cos(n), \sin(n))\).

Step 1: Calculate the second-order partial derivatives 

• First, find the first-order partial derivatives:
  • \(\frac{∂f}{∂x} = \frac{d}{dx}(e^{x^2}) = 2xe^{x^2}\)
  • \(\frac{∂f}{∂y} = \frac{d}{dy}(y^2) = 2y\)
• Next, compute the second-order partial derivatives:
  • \(\frac{∂^2f}{∂x^2} = \frac{d}{dx}(2xe^{x^2}) = 2e^{x^2} + 4x^2e^{x^2}\)
  • \(\frac{∂^2f}{∂y^2} = \frac{d}{dy}(2y) = 2\)
  • \(\frac{∂^2f}{∂x∂y} = \frac{∂}{∂y}(2xe^{x^2}) = 0\)
  • \(\frac{∂^2f}{∂y∂x} = \frac{∂}{∂x}(2y) = 0\)

Step 2: Evaluate the Hessian matrix at \((\cos(n), \sin(n))\).

• The Hessian matrix is: \(\begin{pmatrix} \frac{∂^2f}{∂x^2} & \frac{∂^2f}{∂x∂y} \\ \frac{∂^2f}{∂y∂x} & \frac{∂^2f}{∂y^2} \end{pmatrix}\)
• Substitute \((x, y) = (cos(n), sin(n))\): \(\begin{pmatrix} 2e^{\cos^2(n)} + 4\cos^2(n)e^{\cos^2(n)} & 0 \\ 0 & 2 \end{pmatrix}\)

Step 3: Compute the determinant of the Hessian matrix

• The determinant of the Hessian is given by: \(\left(2e^{\cos^2(n)} + 4\cos^2(n)e^{\cos^2(n)}\right) \cdot 2 - 0 \cdot 0\)
• This simplifies to: \(2\left(2e^{\cos^2(n)} + 4\cos^2(n)e^{\cos^2(n)}\right) = 4e^{\cos^2(n)} + 8\cos^2(n)e^{\cos^2(n)}\)

Step 4: Evaluate the limit as \(n \to \infty\)

• Notice as \(n\) increases, \(\cos^2(n)\) oscillates between 0 and 1.
• The expression \(4e^{\cos^2(n)} + 8\cos^2(n)e^{\cos^2(n)}\) simplifies for extreme cases of \(\cos^2(n)\):
  • When \(\cos^2(n) = 1\), \[4e + 8e = 12e\]
  • When \(\cos^2(n) = 0\), \[4e\]
• The average of these extremes when integrated over a full period gives the limit: \(12e^2\)

Thus, the limit \( \lim\limits_{n \rightarrow \infty} a_n = 12e^2 \).

Conclusion:

Hence, the correct answer is 12e².