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Critical points in calculus are specific values of a function where its derivative is zero or undefined. These points are essential for determining the behavior of functions, such as whether they are increasing or decreasing.
To find the critical points of a function like \(f(x) = 4x + \frac{9}{x}\), we first need to determine the derivative. A critical point occurs where \(f'(x) = 0\) or where the derivative is undefined.
For the given function:

• The derivative \(f'(x) = 4 - \frac{9}{x^2}\) is set to 0, leading to \(4 = \frac{9}{x^2}\).
• Solving gives critical points at \(x = \pm \frac{3}{2}\), and observe the behavior or continuity of the derivative, noting it is undefined at \(x = 0\).

By identifying critical points, we can begin to investigate how the function behaves around these values and determine the nature of these points in terms of maxima and minima.

The intervals at which a function is increasing or decreasing are determined by the sign of its derivative. When \(f'(x) > 0\), the function is increasing, and when \(f'(x) < 0\), the function is decreasing.
For the function \(f(x) = 4x + \frac{9}{x}\), we have:

• The critical points \(x = \pm \frac{3}{2}\) split the number line into intervals:
  • \((-\infty, -\frac{3}{2})\)
  • \((-\frac{3}{2}, \frac{3}{2})\)
  • \((\frac{3}{2}, \,\infty)\)
• Choosing test points in each interval (e.g., \(x = -2, 0, 2\)), we determine:
  • \(f'(-2) = \frac{7}{4} > 0\)
  • \(f'(0)\) does not exist, indicating a possible undefined interval yet with behavior to analyze near it.
  • \(f'(2) = \frac{7}{4} > 0\)

These calculations show that the function is increasing on all the intervals defined except where undefined at \(x = 0\). This analysis is useful for sketching the general shape of the graph.

Local maxima and minima represent the highest or lowest points in a small section of a graph. These can be found using the First Derivative Test.
For the function given, after finding the critical points, we utilize the first derivative test to determine if these points are local maxima or minima:

• At \(x = -\frac{3}{2}\), since \(f'(x)\) does not change sign from positive to negative, there is neither a local maximum nor minimum at this point.
• At \(x = \frac{3}{2}\), \(f'(x)\) remains positive, indicating no local maximum or minimum either.

The absence of sign changes in the derivative across these points tells us that the function does not have local maxima or minima at these critical values, but simply continues increasing through them.

The derivative of a function gives us a way to quickly find the slope of the function at any point. This is critical for understanding how the function behaves over different intervals. The derivative can indicate if the function is increasing or decreasing.
For the function \(f(x) = 4x + \frac{9}{x}\), its derivative was calculated as:

• Derivative: \(f'(x) = 4 - \frac{9}{x^2}\)

This expression helps us analyze the slope of the tangent lines to the function at any given point. Applying derivative rules effectively helps parse out the critical points and test intervals for increasing or decreasing behavior.
Moreover, understanding this derivative form reveals where any undefined points may exist, such as \(x = 0\) in this instance, which is crucial for assessing the function's continuity and breaks.