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An important idea in calculus is determining whether a function is increasing or decreasing over specific intervals. A function is said to be increasing on an interval if, as the input variable (often denoted as \(x\)) increases within that interval, the function's output also rises. Conversely, it is considered decreasing if the function’s output falls as \(x\) rises within that interval.

To find these intervals, you typically examine the sign of the derivative of the function. If the derivative is positive over an interval, the function is increasing there. If it's negative, the function is decreasing.

• Positive Derivative: The function is increasing.
• Negative Derivative: The function is decreasing.

Calculating these intervals helps in sketching the graph of the function and understanding its overall behavior.

Critical points are essential in identifying where a function's behavior changes from increasing to decreasing or vice versa. A critical point occurs where the derivative of the function is zero or undefined. This means we look for points where the slope of the tangent to the curve becomes flat.

To find critical points:

• Calculate the derivative of the function.
• Set the derivative equal to zero and solve for \(x\).

In our exercise, we set the derivative \(y' = 1 - \frac{9}{x^2}\) equal to zero and found the critical points at \(x = 3\) and \(x = -3\). These points help us break down the domain into intervals where the function could be increasing or decreasing.

Derivatives play a key role in calculus, representing the rate of change of a function. In simple terms, the derivative measures how a function's value changes as its input changes slightly.

For the given function \(y = x + \frac{9}{x}\), the derivative is \(y' = 1 - \frac{9}{x^2}\). This derivative was calculated using two basic rules of differentiation:

• Power Rule: Used for terms like \(x\).
• Quotient Rule: Tackles the term \(\frac{9}{x}\).

Derivatives help us uncover vital information about the behavior of functions, such as detecting intervals of increase or decrease, and identifying critical points.

Intervals are the specific sections of the \(x\)-axis over which we analyze a function's behavior. With respect to increasing or decreasing functions, identifying these intervals allows us to delineate where a function grows or declines. By breaking down the function according to critical points, we look into particular segments of the domain.

In our example, after determining the critical points \(x=3\) and \(x=-3\), we established three intervals: \((-\infty, -3)\), \((-3, 3)\), and \((3, \infty)\). By picking test points from each interval and plugging these into the derivative, you can determine where the function increases or decreases. This method helps understand not just isolated points, but entire sections of the function's graph, giving insight into the big picture of its behavior.

• The function is increasing for \(x < -3\) and \(x > 3\) because the derivative is positive.
• The function is decreasing between \(x = -3\) and \(x = 3\) where the derivative is negative.