We are given the function $f(x, y) = 5x^2y + 8xy^3$.

To calculate the second-order partial derivatives, we'll follow these steps:

1. First-order partial derivatives:

• Partial derivative with respect to $x$: $f_x(x, y) = \frac{\partial}{\partial x}(5x^2y + 8xy^3) = 10xy + 8y^3$

• Partial derivative with respect to $y$: $f_y(x, y) = \frac{\partial}{\partial y}(5x^2y + 8xy^3) = 5x^2 + 24xy^2$

2. Second-order partial derivatives:

• Second partial derivative with respect to $x$ twice ($f_{xx}(x, y)$): $f_{xx}(x, y) = \frac{\partial}{\partial x}(10xy + 8y^3) = 10y$

• Mixed partial derivative ($f_{xy}(x, y)$): $f_{xy}(x, y) = \frac{\partial}{\partial y}(10xy + 8y^3) = 10x$

• Mixed partial derivative ($f_{yx}(x, y)$): $f_{yx}(x, y) = \frac{\partial}{\partial x}(5x^2 + 24xy^2) = 10x$ Since $f_{xy} = f_{yx}$ (by Clairaut's Theorem on the equality of mixed partial derivatives), this confirms the correctness.

• Second partial derivative with respect to $y$ twice ($f_{yy}(x, y)$): $f_{yy}(x, y) = \frac{\partial}{\partial y}(5x^2 + 24xy^2) = 48xy$

Final results:

$f_{xx}(x, y) = 10y$ $f_{xy}(x, y) = 10x$ $f_{yx}(x, y) = 10x$ $f_{yy}(x, y) = 48xy$

Would you like more details on any of these steps or further explanation?

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Here are 5 related questions to deepen your understanding:

1. What is the geometric interpretation of second-order partial derivatives?
2. How would you interpret the mixed partial derivatives in terms of the curvature of the surface?
3. How do second-order partial derivatives help in determining concavity?
4. What is Clairaut’s theorem, and why does $f_{xy} = f_{yx}$ hold?
5. How can you find critical points using first and second-order partial derivatives?

Tip: When computing second-order derivatives, always verify that the mixed partial derivatives are equal as a consistency check.