We are tasked with finding the minimum value of the function:

$f(x, y, z) = 2x^2 + 2x + y^2 - y + z^2 - z - 3$

subject to the constraint $z = 2y$.

Step 1: Use the constraint $z = 2y$ to eliminate $z$

Substitute $z = 2y$ into the objective function. This gives:

$f(x, y, 2y) = 2x^2 + 2x + y^2 - y + (2y)^2 - 2y - 3$

Simplifying the expression:

$f(x, y, 2y) = 2x^2 + 2x + y^2 - y + 4y^2 - 2y - 3$

$f(x, y, 2y) = 2x^2 + 2x + 5y^2 - 3y - 3$

Now the function is reduced to two variables, $x$ and $y$, i.e., $f(x, y) = 2x^2 + 2x + 5y^2 - 3y - 3$.

Step 2: Minimize $f(x, y)$

To find the minimum, we take partial derivatives with respect to $x$ and $y$ and set them equal to zero.

2.1: Derivative with respect to $x$

$\frac{\partial}{\partial x}(2x^2 + 2x + 5y^2 - 3y - 3) = 4x + 2$

Set this equal to zero:

$4x + 2 = 0$

$x = -\frac{1}{2}$

2.2: Derivative with respect to $y$

$\frac{\partial}{\partial y}(2x^2 + 2x + 5y^2 - 3y - 3) = 10y - 3$

Set this equal to zero:

$10y - 3 = 0$

$y = \frac{3}{10}$

Step 3: Find the corresponding value of $z$

From the constraint $z = 2y$, we substitute $y = \frac{3}{10}$:

$z = 2 \times \frac{3}{10} = \frac{6}{10} = \frac{3}{5}$

Step 4: Compute the minimum value of $f(x, y, z)$

Now that we have $x = -\frac{1}{2}$, $y = \frac{3}{10}$, and $z = \frac{3}{5}$, we substitute these values into the objective function:

$f\left(-\frac{1}{2}, \frac{3}{10}, \frac{3}{5}\right) = 2\left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right) + 5\left(\frac{3}{10}\right)^2 - 3\left(\frac{3}{10}\right) - 3$

Simplify step by step:

$= 2\left(\frac{1}{4}\right) + 2\left(-\frac{1}{2}\right) + 5\left(\frac{9}{100}\right) - 3\left(\frac{3}{10}\right) - 3$ $= \frac{1}{2} - 1 + \frac{45}{100} - \frac{9}{10} - 3$ $= \frac{1}{2} - 1 + \frac{9}{20} - \frac{9}{10} - 3$

Now, combine all terms:

$= \frac{1}{2} - 1 + \frac{9}{20} - \frac{18}{20} - 3$ $= \frac{10}{20} - \frac{20}{20} + \frac{9}{20} - \frac{18}{20} - \frac{60}{20}$ $= \frac{10 - 20 + 9 - 18 - 60}{20}$ $= \frac{-79}{20}$

Thus, the minimum value of the function is:

$f_{\text{min}} = -\frac{79}{20}$

Final Answer:

$f_{\text{min}} = -\frac{79}{20} \approx -3.95$

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Would you like more details on any step or have any other questions?

Here are some related questions you might find interesting:

1. How do we handle constrained optimization with other methods like Lagrange multipliers?
2. What would happen if the constraint were $z = 3y$ instead of $z = 2y$?
3. How do second derivatives help in confirming the nature of the critical point in optimization problems?
4. Can you explain the geometric interpretation of this optimization problem?
5. How would the minimum value change if the function was $f(x, y, z) = x^2 + y^2 + z^2$?

Tip: To avoid mistakes while substituting constraints in optimization problems, always ensure that you correctly replace all instances of the constrained variable in the objective function before differentiating.