Here's a partially-filled Hessian matrix.
$$
\left[\begin{array}{cc}
? ? ? & -x y \sin (x y) \
-x y \sin (x y) & -x^{2} \sin (x y)
\end{array}\right]
$$

What is the missing entry?

Choose 1 answer:
(A) $-y^{2} \sin (x y)$
(B) $y^{2} \sin (x y)$
(C) $2-x^{2} \sin (x y)$
(D) There's not enough information.

Question

Here's a partially-filled Hessian matrix.
$$
\left[\begin{array}{cc}
? ? ? & -x y \sin (x y) \
-x y \sin (x y) & -x^{2} \sin (x y)
\end{array}\right]
$$

What is the missing entry?

Choose 1 answer:
(A) $-y^{2} \sin (x y)$
(B) $y^{2} \sin (x y)$
(C) $2-x^{2} \sin (x y)$
(D) There's not enough information.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the missing entry in the Hessian matrix, we need to calculate the second partial derivative of the function with respect to x. The function given is $ -x^2 \sin(xy) $. ‖ ‖ step 2 ⋮ The first partial derivative of the function with respect to x is $ \frac{\partial}{\partial x}(-x^2 \sin(xy)) = -2x \sin(xy) - x^2 y \cos(xy) $. ‖ ‖ step 3 ⋮ The second partial derivative of the function with respect to x is $ \frac{\partial^2}{\partial x^2}(-x^2 \sin(xy)) = \frac{\partial}{\partial x}(-2x \sin(xy) - x^2 y \cos(xy)) $. ‖ ‖ step 4 ⋮ Simplifying the second partial derivative with respect to x, we get $ -2 \sin(xy) - 2xy \cos(xy) - 2xy \cos(xy) + x^2 y^2 \sin(xy) $. ‖ ‖ step 5 ⋮ Combining like terms, the second partial derivative with respect to x is $ -2 \sin(xy) - 4xy \cos(xy) + x^2 y^2 \sin(xy) $. This is the missing entry in the Hessian matrix. ‖ ***A*** ∻Key Concept∻ ⚹Hessian Matrix and Second Partial Derivatives⚹ ∻Explanation∻ ⚹The Hessian matrix is composed of second-order partial derivatives of a function. The diagonal entries are the second partial derivatives with respect to each variable, which in this case is x for the missing entry.⚹◊What is the formula to find the Hessian matrix entry at the top left position in terms of the given⍭ Generate me a similar question◊

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