The given equation in the image corresponds to a derivative-based theorem, likely the Mean Value Theorem (MVT) in calculus. It states:

$f'(c) = \frac{f(b) - f(a)}{b - a},$

where $c$ is a point in the open interval $(a, b)$ such that the derivative $f'(c)$ equals the average rate of change of the function $f(x)$ over the interval $[a, b]$.

Key Question

The problem asks: What interval does $c$ lie in?

Answer

From the Mean Value Theorem:

• $c$ lies in the open interval $(a, b)$.

Reasoning:

1. The theorem is applicable under two main conditions:

   • $f(x)$ is continuous on the closed interval $[a, b]$.
   • $f(x)$ is differentiable on the open interval $(a, b)$.

2. The conclusion of the theorem guarantees that there exists at least one $c \in (a, b)$ satisfying the equation.

Would you like to see more detailed examples or derivations? Here are some related questions:

Related Questions:

1. What are the conditions for applying the Mean Value Theorem?
2. How is the Mean Value Theorem applied to real-world problems?
3. Can there be more than one $c$ in $(a, b)$ that satisfies $f'(c) = \frac{f(b) - f(a)}{b - a}$?
4. How does the Mean Value Theorem relate to Rolle’s Theorem?
5. What happens if $f(x)$ is not differentiable at some point in $(a, b)$?

Tip:

Whenever solving calculus problems involving intervals, always check the continuity and differentiability of the function on the respective intervals. This helps determine if theorems like MVT or Rolle's are valid.