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The First Derivative Test is a method used to determine whether a critical point is a local maximum or a local minimum. First, let's remember that a critical point is where the derivative of a function is zero or undefined.
To use the First Derivative Test, follow these simplified steps:

• Find the derivative of the function.
• Identify the critical points by setting this derivative to zero.
• Check the sign of the derivative on either side of each critical point.

If the derivative changes from negative to positive at a critical point, you have a local minimum. Conversely, if it changes from positive to negative, you have a local maximum. For the function given, we see that around the critical point of 0, the derivative goes from negative to positive. This tells us that the function has a local minimum there. The First Derivative Test is a powerful tool for quickly determining the nature of a critical point.

The Second Derivative Test provides an alternative method to determine the nature of a critical point. This test is typically simpler because it involves checking the value of the second derivative at the critical point itself.
Here's how to perform the Second Derivative Test:

• Calculate the second derivative of the function.
• Evaluate the second derivative at each critical point.

If the result is positive, the function is concave up at that point, indicating a local minimum. If negative, the function is concave down, indicating a local maximum. However, when the result is zero, the test is inconclusive, and further investigation may be necessary. In this scenario, at the critical point z = 0, the second derivative test is inconclusive because the second derivative is zero. Thus, using only the second derivative test might not be sufficient without additional context.

A local maximum occurs at a critical point where the function value is greater than the values immediately surrounding it. In simpler terms, it's the "peak" of a hill on a graph. Finding a local maximum can be done with derivative tests.
Here's a quick look at what to consider:

• First identify critical points using derivatives.
• Use the First Derivative Test to see if the derivative changes from positive to negative at that point.

The term "local" indicates that the function values are only compared to nearby points, not to the entire graph. In the given function problem, we found no indication of a local maximum at the critical point, as the necessary conditions during the tests weren't satisfied. While this function in focus has a minimum, local maxima can still occur in other functions you might encounter.

A local minimum is the value of the function at a critical point which is lower than the values surrounding it. This point is the "valley" or lowest point in that neighborhood on the graph.
To find a local minimum:

• Use the First Derivative Test to identify if the derivative changes from negative to positive around the critical point.
• The Second Derivative Test can also confirm a local minimum if it is positive, showing concave up behavior.

In the case of our function, using the First Derivative Test, we were able to identify that there is indeed a local minimum at the critical point of z = 0, due to the derivative changing from negative to positive. This is essential in finding how the function behaves around certain points and understanding the topology of its graph.