Finding Relative Maximum and Minimum Points | A Step-by-Step Guide for Mathematical Analysis

To find relative max or min

To find relative maximum and minimum points of a function, we need to follow these steps:

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To find relative maximum and minimum points of a function, we need to follow these steps:

1. Differentiate the function: Take the derivative of the function.

2. Set the derivative equal to zero: Find the values of x that make the derivative equal to zero. These are the potential critical points where the function may have relative maximums or minimums.

3. Solve for x: Solve the equation obtained in step 2 to find the values of x.

4. Find the corresponding y-values: Plug the values of x obtained in step 3 into the original function to find the corresponding y-values.

5. Determine the nature of the critical points: Evaluate the second derivative of the function at the critical points. If the second derivative is positive, the critical point represents a relative minimum. If the second derivative is negative, the critical point represents a relative maximum. If the second derivative is zero or does not exist, the test is inconclusive.

6. Identify the relative maximums and minimums: Compare the y-values obtained in step 4 to determine which values represent relative maximum or minimum points.

7. State the relative maximum and minimum points: Write down the coordinates of the relative maximum and minimum points (x, y).

It is important to note that these points represent local or relative maximums and minimums within a specific range or interval of the function. To find the absolute maximum and minimum points, we need to evaluate the function at its endpoints or on the entire domain.

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