Mean Value Theorem

To calculate the mean deviation about the mean for the given data, we first need to find the mean.
Step 1: Find the midpoints of each class interval. The midpoints of the class intervals are calculated as the average of the lower and upper limits of the intervals.
Step 2: Find the weighted mean. To find the mean , use the formula: where is the frequency and is the midpoint of each class interval. Thus, the mean is:
Step 3: Find the mean deviation. The mean deviation is given by: Now, calculate for each midpoint: Next, calculate the weighted sum of deviations: Thus, the mean deviation is:

Step 1: Expressing the given function
We rewrite: Using the identity: we obtain: Step 2: Finding the critical points
Differentiate : Setting : Using , we substitute: Dividing by (except when ): Solving for in : These give two values in .
Step 3: Verifying maximum values We check or use the first derivative test. It turns out that attains a maximum at these two points.
Step 4: Conclusion Thus, the number of values of where attains its maximum is: