Solved step by step with explanation. - Function Analysis: Partial Derivatives

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Solved step by step with explanation. - Function Analysis: Partial Derivatives

\[

f(x, y) = \begin{cases}

\]

Statement 1: \( f(x) = (x + 2) (2x + 3y + 1) \)

f_x(-2, 1) = \frac{\partial}{\partial x} (x + 2) \bigg|_{(-2, 1)} = 1

\]

Statement 5: \( f_y(-2, 1) \) does not exist

This statement is true. At the point \((-2, 1)\), the function \( f(x, y) \) is defined only by the first case \( x + 2 \), which does not depend on \( y \). Therefore, the partial derivative with respect to \( y \) does not exist at this point.

- Statement 5: \( f_y(-2, 1) \) does not exist

This analysis highlights the importance of understanding the different cases of the function's definition and how partial derivatives are calculated for each case when evaluating the statements.