The problem you've provided involves applying the Mean Value Theorem (MVT). Here’s how we can break it down:

Problem Breakdown:

• You are given a continuous function $f$ on the interval $[a, b]$, where $f(b) > f(a)$.
• The function $f$ is differentiable for all $x$ in the open interval $(a, b)$, except at $x = 0$.
• We are tasked with finding how many values of $c$ (where $a < c < b$) satisfy the equation:

$\lim_{{x \to c}} \frac{f(x) - f(c)}{x - c} = \frac{f(b) - f(a)}{b - a}$

Application of the Mean Value Theorem:

The Mean Value Theorem states that if a function is continuous on a 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:

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

This means that the slope of the tangent line to the graph of $f(x)$ at $x = c$ is equal to the slope of the secant line joining the points $(a, f(a))$ and $(b, f(b))$.

Key Observations from the Graph:

• The function appears to be smooth and differentiable everywhere except at $x = 0$, where there might be a cusp or sharp corner.
• The slope of the secant line between $(a, f(a))$ and $(b, f(b))$ is positive since $f(b) > f(a)$.
• The graph shows multiple turning points, suggesting multiple values of $c$ where the tangent line could have the same slope as the secant line.

How Many Values of $c$?

Looking at the graph, there are potentially three values of $c$ in the interval $(a, b)$ where the tangent line would have the same slope as the secant line. These values are associated with the intervals between the peaks and valleys of the curve.

Thus, the answer is that there are likely three values of $c$ where the condition of the Mean Value Theorem holds.

Would you like further clarification or details? Here are some related questions for exploration:

1. How does the Mean Value Theorem apply to functions that are not differentiable at a point?
2. Can the Mean Value Theorem be applied to functions that are discontinuous?
3. What are the conditions that must be met for the Mean Value Theorem to hold?
4. How would the result change if the function were not differentiable at more points than just $x = 0$?
5. How do the turning points on the graph influence the values of $c$ in the Mean Value Theorem?

Tip: When applying the Mean Value Theorem, always verify that the function is continuous on the closed interval and differentiable on the open interval before applying it.