Mean Value Theorem and graphs Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of f on [1,3]. Reconcile your results with the Mean Value Theorem. <IMAGE>

Verified step by step guidance

1. First, calculate the average rate of change of the function f on the interval [1, 3] using the formula: \( \frac{f(3) - f(1)}{3 - 1} \).
2. Next, find the derivative of the function f, denoted as f'(x), which represents the slope of the tangent line at any point x.
3. Set the derivative f'(x) equal to the average rate of change calculated in step 1 to find the points where the slope of the tangent line equals the average rate of change.
4. Solve the equation from step 3 for x within the interval (1, 3) to find the specific points that satisfy this condition.
5. Finally, verify that the points found in step 4 are indeed within the interval (1, 3) and reconcile these results with the Mean Value Theorem, which states that at least one such point must exist in the interval.