Step 1: Apply Lagrange's Mean Value Theorem (M.V.T.):
The Lagrange's Mean Value Theorem states that for a function continuous on the closed interval and differentiable on the open interval , there exists some such that: In this problem, we are given the function on the interval . We need to find where the conclusion of the M.V.T. holds.
Step 2: Calculate and :
First, we find the values of the function at the endpoints of the interval:
Step 3: Find the average rate of change:
The average rate of change of the function over the interval is:
Step 4: Find the derivative of :
The derivative of the function is:
Step 5: Set the derivative equal to the average rate of change:
According to the M.V.T., there exists some such that: Substituting the expression for :
Step 6: Solve for :
Now, solve for :
Step 7: Verify that :
Since lies in the interval , it is a valid solution.
Conclusion:
Thus, the value of that satisfies the
conclusion of Lagrange’s M.V.T. is .