Finding Critical Numbers | A Step-by-Step Guide to Identify Extrema and Inflection Points

To find critical numbers

To find critical numbers, you need to follow these steps:

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To find critical numbers, you need to follow these steps:

1. Determine the function: Start with the given function for which you want to find the critical numbers. For example, let’s consider the function f(x).

2. Find the derivative: Take the derivative of the function f(x) with respect to x. This will give you a new function, usually denoted as f'(x) or dy/dx. The derivative measures the rate of change of the original function.

3. Set the derivative equal to zero: Solve the equation f'(x) = 0 to find the values of x where the derivative is equal to zero. These values are potential critical numbers because they indicate the points where the original function may have maximum, minimum, or inflection points.

4. Determine if the critical numbers are valid: To verify if the critical numbers found in step 3 are valid, you need to consider both the values of the original function f(x) and the behavior of the derivative f'(x) around these points. Check if the derivative changes sign on either side of the critical numbers.

– If the original function changes from increasing to decreasing (or vice versa) at a critical number, then it is a local extremum (maximum or minimum).
– If the function continues to increase or decrease at the critical number, or if the sign of the derivative does not change, then the critical number may indicate an inflection point or is not relevant.

By following these steps, you can find the critical numbers of a function and determine their significance in terms of extrema and inflection points. It’s important to note that critical numbers can also exist at points where the derivative does not exist, such as sharp or discontinuous edges.

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