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In the realm of calculus, finding the maxima and minima of a function is essential for understanding its behavior. These points mark where the function reaches its highest or lowest value, respectively. They often represent important features in various disciplines, such as optimization problems in economics or engineering. To determine these points, one typically investigates where the derivative of the function equals zero. This is because a slope of zero signifies a turning point, such as a peak or a trough.

Once you find the critical points, it's crucial to check their nature. A helpful way to confirm whether these are maxima or minima is by taking the second derivative. If the second derivative at a point is negative, it's a local maximum; if positive, it's a local minimum. In our function, the second derivative test revealed a negative value, confirming the presence of a local maximum at the found critical point. These techniques ensure a thorough understanding of function behavior across its domain.

Differential calculus is all about the rate of change. By taking derivatives, we get an insight into how a function changes as its variables change. When assessing a function to find its maxima or minima, derivatives guide us in finding those critical points where the function changes direction.

In our exercise, after expressing the given function in terms of a single variable, differential calculus helped find when the slope of the curve became zero. This was achieved by deriving the function with respect to the variable and setting the result equal to zero. The solution focused on this derivative, leading to the discovery of the critical point.

Additionally, by evaluating the second derivative, we gain further insights into the function's concavity, telling us whether we're looking at a peak or a valley. This blend of first and second derivatives offers a powerful toolkit for understanding and optimizing functions, which is frequently applied in countless real-world scenarios.

Functions of several variables are built to handle more complex scenarios than functions with just one variable. In such functions, the idea is to explore how multiple variables interplay to affect outcomes. For example, given a function of the form \( f(x, y, z) = xyz \), each of these variables contributes to the final value of the function.

In this exercise, the function is initially in three variables. By using a parametric representation, we brought it down to a single variable problem where optimization becomes more straightforward. The relationship among multiple variables was simplified, allowing us to tackle it using the intuitive methods of single-variable calculus.

This approach highlights the convenience and necessity of understanding functions in multiple variables, especially when considering constraints or specific paths like lines in space. Such techniques are essential in multivariable calculus, playing a significant role in fields like economics, physics, and computer graphics, where multiple factors must be simultaneously considered.