Question

Math

Posted 4 months ago

```
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.
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

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.

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