To show that there is at least one point between $x = a$ and $x = b$ where the tangents to the graphs of $f$ and $g$ are either parallel or the same line, we can apply the Mean Value Theorem from calculus and leverage the given information.

Solution:

1. Given Information:

   • $f$ and $g$ are differentiable on the interval $[a, b]$.
   • $f(a) = g(a)$ and $f(b) = g(b)$.

2. Construct a New Function: Define a new function $h(x) = f(x) - g(x)$. This function represents the difference between $f(x)$ and $g(x)$ on the interval $[a, b]$.

3. Properties of $h(x)$:

   • Since $f$ and $g$ are differentiable, $h(x)$ is also differentiable on $[a, b]$.
   • At the endpoints, $h(a) = f(a) - g(a) = 0$ and $h(b) = f(b) - g(b) = 0$.

4. Apply Rolle's Theorem:

   • Rolle’s Theorem states that if a function is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and the values at the endpoints are the same (i.e., $h(a) = h(b)$), then there exists at least one point $c \in (a, b)$ such that $h'(c) = 0$.
   • Since $h(a) = h(b) = 0$, we can apply Rolle's Theorem to $h(x)$ on the interval $[a, b]$.

5. Interpret $h'(c) = 0$:

   • $h'(x) = f'(x) - g'(x)$, so $h'(c) = 0$ implies $f'(c) - g'(c) = 0$, or equivalently $f'(c) = g'(c)$.
   • This result means that the derivatives (slopes of the tangents) of $f$ and $g$ at $x = c$ are equal.

6. Conclusion: Since $f'(c) = g'(c)$, the tangents to the graphs of $f$ and $g$ at $x = c$ are parallel. If the functions are identical in form around $x = c$, they may even share the same tangent line at that point.

Thus, we have shown that there exists at least one point $c \in (a, b)$ where the tangents to the graphs of $f$ and $g$ are either parallel or the same line.

Would you like more details on any step?

Here are five related questions that might interest you:

1. What are other applications of Rolle’s Theorem?
2. Can we conclude that $f$ and $g$ are equal on $[a, b]$ if their tangents are parallel at every point?
3. How does the Mean Value Theorem generalize Rolle's Theorem?
4. What conditions are necessary for Rolle's Theorem to apply?
5. How would this proof change if $f$ and $g$ were only continuous on $[a, b]$ but not differentiable?

Tip: Rolle’s Theorem is a specific case of the Mean Value Theorem, and it’s useful in proving that a function has at least one stationary point on an interval.