Let's solve the given problem step by step.

The function is:
$f(x, y) = 4x^2 + 6y^2 - 10$

You are tasked to find $f_{xx}(x, y)$, which is the second partial derivative of $f(x, y)$ with respect to $x$.

Step 1: First Partial Derivative with respect to $x$

$f_x(x, y) = \frac{\partial}{\partial x}(4x^2 + 6y^2 - 10)$

Since $4x^2$ is dependent on $x$, and $6y^2 - 10$ are constants with respect to $x$, we get: $f_x(x, y) = 8x$

Step 2: Second Partial Derivative with respect to $x$

Now, take the derivative of $f_x(x, y) = 8x$ with respect to $x$: $f_{xx}(x, y) = \frac{\partial}{\partial x}(8x) = 8$

Final Answer:

The value of $f_{xx}(x, y)$ is 8.

Thus, the correct option is: $\textbf{Option 4: 8}$

Would you like me to clarify any part of the solution?
Here are 5 related questions to expand your understanding:

1. What is the geometric meaning of second partial derivatives in multivariable functions?
2. How do you calculate mixed partial derivatives like $f_{xy}$?
3. What happens when you take higher-order partial derivatives of a polynomial function?
4. Can you apply the same process to a non-polynomial function, such as $e^x$ or $\sin(x)$?
5. How do second partial derivatives relate to concavity and critical points?

Tip:

When dealing with partial derivatives, always pay attention to which variable you're differentiating with respect to, and treat the other variables as constants!