This problem requires us to find the point on the curve where the instantaneous rate of change (slope of the tangent) is equal to the average rate of change (slope of the secant) over the interval $[a, b]$. This concept relates to the Mean Value Theorem (MVT) for derivatives, which states that if a function $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one point $c$ in $(a, b)$ such that:

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

Solution Process:

• Average rate of change: This is the slope of the line connecting the points $(a, f(a))$ and $(b, f(b))$. In the given graph, it corresponds to the slope of the secant line through points $A$ and $D$.

• Instantaneous rate of change: At a point on the curve, this is the slope of the tangent line at that point. We need to find a point on the graph where the tangent slope matches the secant slope between $A$ and $D$.

• Inspection of the Graph:

  1. The secant slope between $A$ (at $x = a$) and $D$ (at $x = b$) is negative since the curve decreases overall between these points.
  2. We are looking for a point between $a < x < b$ where the slope of the tangent line is equal to this negative secant slope.

• Checking Tangent Slopes at Labeled Points:

  • At A: The slope of the tangent is zero (horizontal line).
  • At B: The slope is slightly negative.
  • At C: The slope appears most likely to match the negative secant slope, as the tangent at $C$ looks similar in steepness to the average slope between $A$ and $D$.
  • At D: The slope is also zero (horizontal line).

Conclusion:

The correct answer is likely C, as this is the point where the instantaneous slope (tangent) most likely matches the average slope (secant) between $A$ and $D$.

Do you have any questions or need further details?

Here are 5 related questions for further exploration:

1. What is the Mean Value Theorem, and how is it applied in calculus?
2. How can you determine the slope of a tangent line at a given point?
3. What are other applications of the Mean Value Theorem?
4. What distinguishes average rate of change from instantaneous rate of change?
5. How can one visually estimate slopes on a graph?

Tip: In some cases, practicing problems on the Mean Value Theorem helps in recognizing slopes and tangents intuitively.