91Ó°ÊÓ

In the context of calculus, critical points are pivotal in understanding the behavior of a function. They are essentially the 'crossroads' of a graph where the function's rate of change shifts gears. Specifically, a critical point occurs where the derivative of a function equals zero or the derivative does not exist.

For the given function, \(h(x) = 12 x - x^3\), finding the critical points involved setting the derivative \(h'(x) = 12 - 3x^2\) to zero. As the derivative is a continuous, well-defined function across all real numbers, we did not have critical points where the derivative does not exist. Solving \(12 - 3x^2 = 0\) returns the values \(x = -2\) and \(x = 2\), which are the crucial positions where the slope of the tangent to the curve is zero—meaning the graph has a potential for peaks or troughs at these points.

When dissecting functions and their behavior, the derivative of a function is an indispensable tool. Mathematically, the derivative represents the rate of change or the slope of the curve of a function at any given point. Its fundamental importance lies in its ability to predict the 'motion' of the function's graph—whether it's rising or falling as you move along the x-axis.

To calculate the derivative of \(h(x) = 12x - x^3\), we applied the power rule separately to each term of the function. The derivative, \(h'(x) = 12 - 3x^2\), gave us a new function that reports the gradient of our initial function at any point \(x\). If \(h'(x) > 0\), the function \(h\) is increasing at that point, and if \(h'(x) < 0\), it is decreasing. The derivative is the primary stepping-stone to identifying the intervals of increase and decrease, crucial for sketching the shape of the function.

The power rule is a straightforward yet powerful derivative rule in calculus. It's essentially a quick method to find the derivative of any function where the variable \(x\) is raised to a power. The rule states that if \(f(x) = x^n\), where \(n\) is any real number, then the derivative \(f'(x) = nx^{n-1}\). No matter if \(n\) is positive, negative, or zero, this rule applies.

In our example, taking the derivative of \(12x\) and \(x^3\) using the power rule resulted in \(h'(x) = 12 - 3x^2\). Each term of the original function was treated separately: the exponent in \(12x\) is implicitly 1, so it became 12 (since 1 times 12 is 12 and reducing the exponent by 1 gives \(x^0\), which is 1), and for \(x^3\), the derivative became \(3x^2\). This rule provided a quick solution to finding the function's derivative, an essential step for analyzing its increasing and decreasing behavior.

Once critical points are identified, the intervals surrounding these points need to be tested to see where the function is increasing or decreasing—a process we refer to as the test intervals method. This method involves picking sample points (test points) from each interval created by the critical points and evaluating the derivative at these points.

In the given function, the critical points split the real line into three intervals: \( (-\infty, -2), (-2, 2)\), and \( (2, \infty)\). By testing points from each interval—like \(x = -3, 0, 3\)—we plugged them into \(h'(x)\). The signs of the derivative's outputs indicated the function's behavior: positive meant the function was increasing, and negative implied it was decreasing. From the calculations, \(h(x)\) is increasing on \( (-\infty, -2)\) and \( (-2, 2)\) but decreasing on \( (2, \infty)\).