Examine the following functions for maxima and minima:

a.

b.

To examine the given functions for maxima and minima, we need to use the method of partial derivatives and check the second-order partial derivatives for critical points. We will analyze each function step by step and compute necessary derivatives to apply the conditions for extrema (maxima or minima).

### a) \( z = 2x^2 + xy + 4y^2 + xz + z^2 + 2 \)

#### Step 1: Compute the partial derivatives
We first compute the first-order partial derivatives with respect to \( x \), \( y \), and \( z \), as these will help us locate the critical points.

1. Partial derivative with respect to \( x \):
   \[
   \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (2x^2 + xy + 4y^2 + xz + z^2 + 2)
   \]
   Differentiate each term:
   \[
   \frac{\partial z}{\partial x} = 4x + y + z
   \]

2. Partial derivative with respect to \( y \):
   \[
   \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (2x^2 + xy + 4y^2 + xz + z^2 + 2)
   \]
   Differentiate each term:
   \[
   \frac{\partial z}{\partial y} = x + 8y
   \]

3. Partial derivative with respect to \( z \):
   \[
   \frac{\partial z}{\partial z} = \frac{\partial}{\partial z} (2x^2 + xy + 4y^2 + xz + z^2 + 2)
   \]
   Differentiate each term:
   \[
   \frac{\partial z}{\partial z} = x + 2z
   \]

#### Step 2: Set the partial derivatives equal to zero (Critical points)
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