Existence of Points c1 and c2

Given a differentiable function f: [a, b] → R, we need to determine the truth of the statement "there exist points c1 and c2 in the interval (a, b)."

Mean Value Theorem

To prove or disprove the statement, we can make use of the Mean Value Theorem. The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the open interval (a, b) such that the derivative of f at c is equal to the average rate of change of f over [a, b].

Proof

Let's assume that the statement is false, i.e., there are no points c1 and c2 in the open interval (a, b). This would mean that the function f is not differentiable at any point in (a, b).

Since f is a differentiable function on the closed interval [a, b], it must be continuous on the closed interval [a, b] as well. By the Extreme Value Theorem, f must attain its maximum and minimum values on the closed interval [a, b]. Let's assume the maximum value of f is attained at point d in [a, b].

If f is not differentiable at d, then it cannot satisfy the conditions of the Mean Value Theorem. However, this contradicts the fact that f is differentiable on the open interval (a, b). Therefore, f must be differentiable at the point of its maximum value.

Similarly, we can argue for the minimum value of f on the closed interval [a, b]. Therefore, we conclude that there exist points c1 and c2 in the open interval (a, b) where the function f is differentiable.

Conclusion

Hence, the statement "there exist points c1 and c2 in the open interval (a, b)" is true.