**Rolle's Theorem**

Rolle's theorem is a fundamental theorem in calculus that states the conditions under which a differentiable function will have a critical point between two points on its graph. The theorem is named after Michel Rolle, a French mathematician who first stated it in the late 17th century.

**Statement of Rolle's Theorem**

Rolle's theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one number c in the open interval (a, b) such that f'(c) = 0.

**Integrating f(x) from 3 to 5**

In this problem, we are given that the function f(x) satisfies the conditions for Rolle's theorem in the interval (3, 5). Let's denote the integral of f(x) from 3 to 5 as ∫[3,5] f(x) dx.

By the fundamental theorem of calculus, if F(x) is an antiderivative of f(x), then the integral of f(x) from a to b is given by F(b) - F(a). Therefore, to find ∫[3,5] f(x) dx, we need to find an antiderivative of f(x).

**Using Rolle's Theorem to Find an Antiderivative**

Since f(x) satisfies the conditions for Rolle's theorem in the interval (3, 5), we know that there exists at least one number c in the open interval (3, 5) such that f'(c) = 0. This means that f(x) has a critical point at x = c.

A critical point of a function occurs when its derivative is equal to zero. Therefore, f'(c) = 0 implies that f(x) has a maximum, minimum, or an inflection point at x = c.

**Determining the Antiderivative of f(x)**

Now, since f(x) has a critical point at x = c, the derivative of f(x) must change sign at x = c. This implies that f'(x) is positive for x < c="" and="" negative="" for="" x="" /> c, or vice versa.

By the mean value theorem, if f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number d in the open interval (a, b) such that f'(d) = (f(b) - f(a))/(b - a).

Since f(a) = f(b) (as mentioned in the statement of Rolle's theorem), we have f'(d) = 0. This means that f(x) has a critical point at x = d.

**Conclusion**

In conclusion, if f(x) satisfies the conditions for Rolle's theorem in the interval (3, 5), then there exists at least one number c in the open interval (3, 5) such that f'(c) = 0. This implies that f(x) has a critical point at x = c.

To find the integral of f(x) from 3 to 5, we need to find an antiderivative of f(x). Since f(x) has a critical point at x = c, the