Let's break down the problem step by step:

1. We are given the function $g(x, y, z) = z^3 + xz + y$.
2. We have limits as $z \to +\infty$ and $z \to -\infty$, showing that $g(x, y, z) \to \infty$ and $g(x, y, z) \to -\infty$ respectively.
3. This suggests that by the Intermediate Value Theorem, the equation $g(x, y, z) = 0$ has a solution for each $x$ and $y$.

Differentiation Process:

• We are asked to find the linear approximation of $f(x, y)$ at $(14, -624)$ using implicit differentiation. This means we treat $z$ as a function of $x$ and $y$ and differentiate both sides of the equation $g(x, y, z) = 0$.

  First, we differentiate implicitly: $\frac{\partial g}{\partial x} = \frac{d}{dx}(z^3 + xz + y) = 0$ Applying the chain rule: $3z^2 \frac{dz}{dx} + z + x \frac{dz}{dx} = 0$ Solving for $\frac{dz}{dx}$: $\frac{dz}{dx} = \frac{-z}{3z^2 + x}$

  Similarly, for $\frac{\partial g}{\partial y}$: $\frac{d}{dy}(z^3 + xz + y) = 0$ Since the only term involving $y$ is linear: $\frac{dz}{dy} = \frac{-1}{3z^2 + x}$

4. Given that $g(14, -624, 8) = 0$, we can use these values for $x = 14$, $y = -624$, and $z = 8$ to evaluate the derivatives:

   $\frac{dz}{dx} = \frac{-8}{3(8)^2 + 14} = \frac{-8}{3(64) + 14} = \frac{-8}{192 + 14} = \frac{-8}{206} \approx -0.0388$ $\frac{dz}{dy} = \frac{-1}{3(8)^2 + 14} = \frac{-1}{206} \approx -0.00485$

5. Using the point $(14, -624)$, the linear approximation formula is: $f(x, y) \approx f(14, -624) + \frac{dz}{dx} (x - 14) + \frac{dz}{dy} (y + 624)$ Since $f(14, -624) = 8$, the linear approximation becomes: $f(x, y) \approx 8 - 0.0388(x - 14) - 0.00485(y + 624)$

Root Approximation:

Finally, you're asked to use this approximation to estimate the root of the equation $z^3 + 19z - 640 = 0$. This can be done by solving iteratively, but the first approximation would be based on $f(x, y) \approx 8$ as a starting point.

Do you want more details or clarification on any of the steps? Here are five relative questions for further exploration:

1. How does the Intermediate Value Theorem apply in multivariable contexts like this one?
2. Why is implicit differentiation necessary in this problem?
3. Can you find the exact root of $z^3 + 19z - 640 = 0$ using the approximation?
4. What other approximation methods could be used to solve the cubic equation?
5. How do changes in $x$ and $y$ influence the solution $z$ in $g(x, y, z) = 0$?

Tip: When performing implicit differentiation, always apply the chain rule carefully to each variable that depends on the others.