Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 65

See explanation below.

Step 1. With the given conditions, we have $f'(x)=g'(x)$ for all $x$ in the domain. Step 2. Using Corollary 2 with the above condition, there exists a constant $C$ such that $f(x)=g(x)+C$. Step 3. Since the two functions start at the same point $a$, we have $f)a)=g(a)$, which means $C=0$. Step 4. We conclude that the two functions are identical: $f(x)=g(x)$