Mean

Step 1: Lagrange's Mean Value Theorem states that if is continuous on the closed interval and differentiable on the open interval , then there exists at least one such that: Step 2: For the given function , it is continuous and differentiable on because it is a polynomial function. Step 3: The interval is , so we compute and : Step 4: Using Lagrange’s mean value theorem: Step 5: Now, compute . The derivative of is: Step 6: Set and solve for : Step 7: Thus, , and we have verified that the Lagrange Mean Value Theorem holds for the function on the interval .

The mean of the numbers from 1 to 15 is given by the formula: The sum of the numbers from 1 to 15 is the sum of an arithmetic series: where , , and . So, The number of numbers is 15. Therefore, the mean is: Thus, the mean is: