Referring to the Mean Value Theorem and Rolle's Theorem, how can I tell if f is continuous on the interval [a,b] and differentiable on (a,b).

To determine if a function f is continuous on the interval [a, b] and differentiable on (a, b), you can use the Mean Value Theorem and Rolle's Theorem. Here's how you can apply each theorem:

1. Mean Value Theorem:
According to the Mean Value Theorem, 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 (a, b) such that the derivative of f at c is equal to the average rate of change of f over the interval [a, b].

To check if the function f satisfies the conditions of the Mean Value Theorem, follow these steps:

Step 1: Check Continuity:
Verify if f is continuous on the closed interval [a, b]. This involves ensuring that f is defined and has no discontinuities such , jumps, or vertical asymptotes within [a, b]. You can examine the graph of f or check for any potential disruptions in the function's domain.

Step 2: Check Differentiability:
Determine if f is differentiable on the open interval (a, b). This means that the derivative of f exists at every point within (a, b). You can check differentiability by computing the derivative of f using any applicable differentiation rules and verifying if it exists for all x in (a, b).

2. Rolle's Theorem:
Rolle's Theorem is a special case of the Mean Value Theorem. It states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where the derivative of f at c is equal to zero.

To check if the function f satisfies the conditions of Rolle's Theorem, you need to go through the following steps:

Step 1: Check Continuity:
Verify if f is continuous on the closed interval [a, b] using the same procedure mentioned earlier.

Step 2: Check Differentiability:
Determine if f is differentiable on the open interval (a, b) as explained previously.

Step 3: Check Endpoint Values:
Verify if f(a) = f(b). This entails evaluating the function at the endpoints a and b and comparing the results. If f(a) = f(b), then the condition of Rolle's Theorem is satisfied.

By following these steps, you can determine if a function f is continuous on the interval [a, b] and differentiable on (a, b) by making use of the Mean Value Theorem and Rolle's Theorem.