Calculate all four second-order partial derivatives of f(x,y)=5x2y+8xy3.

fxx(x,y)= 

fxy(x,y)= 

fyx(x,y)= 

fyy(x,y)= 

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.

Calculate all four second-order partial derivatives of f(x,y)=5x^2y+8xy^3.

fxx(x,y)=

fxy(x,y)=

fyx(x,y)=

fyy(x,y)=

Learn how to compute all four second-order partial derivatives of the function f(x, y) = 5x^2y + 8xy^3. This step-by-step guide explains the process, including the use of Clairaut's Theorem and calculating mixed partial derivatives. Suitable for students studying multivariable calculus.

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