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In the context of graph sketching, a critical point is a point on the graph of a function where the derivative is either zero or undefined. Critical points are important because they indicate potential turning points in the graph, such as maxima, minima, or inflection points.
In our exercise, the function's derivative, denoted as \(f'(x)\), equals zero at \(x = -1\). This marks \(x = -1\) as a critical point.
To identify whether this critical point is a maximum, minimum, or inflection involves further analysis, such as using the first derivative test. In this case, given the information that the function's derivative changes from positive to negative at \(x = -1\), this critical point is determined to be a local maximum.
Recognizing these points is crucial when sketching graphs because they give vital information about the behavior of the function and help in interpreting the function's graph accurately.
Remember:

• Critical points occur where \( f'(x) = 0\) or \( f'(x)\) is undefined.
• They help to identify potential turning points on a graph.

When sketching a graph, understanding where a function is increasing or decreasing is key. A function is said to be increasing on an interval if its derivative is positive over that interval. Conversely, it is decreasing if the derivative is negative.
From the problem scenario:

• For \(x < -1\), \(f'(x) > 0\) indicates the function is increasing.
• For \(x > -1\), \(f'(x) < 0\) implies the function is decreasing.

This change highlights how the function behaves before and after the critical point at \(x = -1\). Graphically, this translates to the function rising towards \(x = -1\) and then falling after passing \(x = -1\).
Recognizing intervals of increase and decrease allows us to construct the graph's overall shape and predict where it might peak or valley. In this exercise, the function creates a peak at \(x = -1\).
Key points to keep in mind:

• A positive derivative signifies an increasing function.
• A negative derivative denotes a decreasing function.

The first derivative test is a useful method for determining the nature of critical points on a graph of a function. It is commonly used to decide whether a critical point is a local maximum, local minimum, or neither.
In our specific case, we used the first derivative test to examine the function around \(x = -1\). Since:

• The function transitions from increasing (\(f'(x) > 0\)) to decreasing (\(f'(x) < 0\)) across the point \(x = -1\).
• We can conclude that there is a local maximum at \(x = -1\).

This result is consistent with the rules of the first derivative test:

• If \(f'(x)\) changes from positive to negative, the function has a local maximum at that point.
• If it changes from negative to positive, it indicates a local minimum.
• If there is no sign change, the critical point may not be a maximum or minimum.

The first derivative test is a powerful tool to add clarity when sketching the overall shape of functions by identifying where peaks and valleys might occur.