Statement:
Let f: [a, b] → R be a differentiable function. There exist points c1, c2 € (a, b).

To Find:
Which of the following is true?

(a) 3f(c1) f(c1)f(c2) [f(a) - f(b)]
(b) 4f(c1) f(c1)=f'(c2) [f(a) f(b)]
(c) 5f(ci) f'(ci) = f(c2) [f(a) f(b)]
(d) 2f(c1) f(c1) = f(c2) [f(a) f(b)]

Solution:
To determine which statement is true, we need to analyze each option.

Option (a):
3f(c1) f(c1)f(c2) [f(a) - f(b)]

This statement does not involve derivatives, so it cannot be true based on the given conditions. We can eliminate this option.

Option (b):
4f(c1) f(c1)=f'(c2) [f(a) f(b)]

The left-hand side of the equation involves the function f(c1), while the right-hand side involves the derivative f'(c2). Since f(c1) and f'(c2) are not directly related, this statement cannot be true. We can eliminate this option.

Option (c):
5f(ci) f'(ci) = f(c2) [f(a) f(b)]

This statement involves the function f(ci) and its derivative f'(ci). However, there is no direct relationship between f(ci) and f'(ci) given in the problem. Therefore, this statement cannot be true. We can eliminate this option.

Option (d):
2f(c1) f(c1) = f(c2) [f(a) f(b)]

This statement involves the function f(c1) and f(c2). Since both c1 and c2 are given as points in the interval (a, b), this statement is plausible. We need to further analyze whether this statement holds true.

Proof:
Let's consider the mean value theorem for differentiation, which states that for any function f(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists a point c in (a, b) such that:
f'(c) = [f(b) - f(a)] / (b - a)

Since f(x) is differentiable on the interval [a, b], it is also continuous on this interval. By applying the mean value theorem, we can find a point c in (a, b) such that:
f'(c) = [f(c2) - f(c1)] / (c2 - c1)

Now, rearranging the equation, we get:
f(c2) - f(c1) = f'(c) * (c2 - c1)

Multiplying both sides of the equation by 2, we have:
2f(c1) * (f(c2) - f(c1)) = 2f(c1) * f'(c) * (c2 - c1)

Since c1