https://r2.erweima.ai/mathbot/image/1723859206811kkce3hpx.jpg

To determine if the functions \( f(x, y) \) defined in your image are convex, we can follow a systematic approach. Here’s a step-by-step outline:

### Step 1: Determine the Domain
First, check the domain of each function to ensure that it is well-defined for real values of \( x \) and \( y \).

### Step 2: Calculate the Hessian Matrix
To determine the convexity of a function \( f(x, y) \), you need to check the Hessian matrix \( H \), which is the matrix of second-order partial derivatives:

\[
H = \begin{pmatrix}
\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\
\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2}
\end{pmatrix}
\]

For the function to be convex, the Hessian matrix must be positive semidefinite for all \( (x, y) \) in its domain.

### Step 3: Check the Determinant of the Hessian Matrix
The determinant of the Hessian matrix can help in verifying whether the matrix is positive semidefinite. Specifically, a function is convex if:
- \( \frac{\partial^2 f}{\partial x^2} \geq 0 \) and \( \text{det}(H) \geq 0 \).

### Step 4: Analyze the Results
If the Hessian matrix is positive semidefinite everywhere, the function is convex over its domain.

Now, I will proceed by analyzing each function using the steps mentioned:

1. **Function 1**: \( f(x, y) = \sqrt{x} \cdot \tan(y) \)
2. **Function 2**: \( f(x, y) = \sqrt{\frac{x}{y}} \cdot \ln(x^2 + y^2 - 4) \)
3. **Function 3**: \( f(x, y) = \sqrt{y} \cdot \ln(x) \)
4. **Function 4**: \( f(x, y) = \text{arcsec}\left(\sqrt{|x| + |y|} - 2\right) + \ln(9 - x^2 - y^2) \)
5. **Function 5**: \( f(x, y) = \ln(x^2 - y) + \sqrt{y^2 + x} \)

### Summary of Expected Outcomes:
1. **Function 1**: Convexity might depend on the signs and ranges of \( x \) and \( y \). The square root is concave, and \( \tan(y) \) is periodic and non-convex.
2. **Function 2**: The logarithmic term might affect convexity, especially since it only takes positive values inside its domain.
3. **Function 3**: Similar concerns about the logarithmic function, which is concave.
4. **Function 4**: The arcsecant function has branches and is not convex, especially combined with a logarithm.
5. **Function 5**: This function has non-linear components (logarithmic and square root), suggesting it might not be convex.

Let me compute these systematically. Would you like the detailed calculations for each step, or just the final conclusion about whether each function is convex?

Learn how to determine if functions are convex using Hessian matrix analysis. Step-by-step guide with examples including functions involving square roots, logarithms, and trigonometric functions.

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