Mastering The First Derivative Test: Finding Relative Maximums And Minimums Of Functions

Relative Min ( 2nd derivative test)

f'(x)=0f(x) > 0Backwards one

The second derivative test is a tool for finding the relative extremum points of a function. Specifically, the test helps in identifying whether a given point is a local maximum or a local minimum. Here, we’re interested in determining the points at which the function turns from concave downwards to concave upwards or vice versa. These points are known as inflection points.

To apply the second derivative test to a function, we need to follow these steps:

Step 1: Find the second derivative of the function f.

Step 2: Find the points where the second derivative changes sign, that is the points where the second derivative is equal to zero or undefined.

Step 3: For each of these points, determine the nature of the extremum.

If the second derivative is positive at the point in question, then the function is concave upwards, and the point corresponds to a local minimum. Conversely, if the second derivative is negative, then the function is concave downwards, and the point corresponds to a local maximum.

Note that the second derivative test doesn’t always give us enough information to determine whether a critical point is a maximum or a minimum. In such cases, we need to apply other methods, such as the first derivative test or using the function’s graph to find more information about its behavior.

So, in summary, the second derivative test is an essential tool for determining the nature of local extrema and inflection points of functions.

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