The problem shown in the image is asking to find at which point on the graph of the trigonometric function $f$ the instantaneous rate of change (i.e., the slope of the tangent) equals the average rate of change of $f$ over the interval $[a, b]$. This is essentially an application of the Mean Value Theorem (MVT), which 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:

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

Step-by-Step Breakdown:

1. Average rate of change: This is the slope of the secant line between the points $(a, f(a))$ and $(b, f(b))$. Visually, this is the slope of the straight line connecting point A to point D on the graph.

2. Instantaneous rate of change: This is the slope of the tangent line at any point on the graph. According to the MVT, we are looking for a point where the tangent slope equals the secant slope.

3. Graph Analysis:

   • Point A: The tangent line at A appears to be horizontal (slope = 0).
   • Point B: The tangent line at B seems to have a negative slope that is similar to the slope of the secant line from A to D.
   • Point C: The slope at C is also negative but appears steeper than the secant slope.
   • Point D: The tangent line at D is horizontal (slope = 0), similar to point A.

Conclusion:

By analyzing the tangent slopes, point $B$ is where the slope of the tangent (instantaneous rate of change) most closely matches the slope of the secant line, thus satisfying the Mean Value Theorem. Therefore, the correct answer is B.

Let me know if you'd like further details or clarification on this problem.
Here are some additional related questions for practice:

1. What conditions must be satisfied for the Mean Value Theorem to apply to a function?
2. How do you calculate the average rate of change of a function over a given interval?
3. What is the difference between the secant line and the tangent line on a function's graph?
4. Can a function have more than one point where the instantaneous rate of change equals the average rate of change? Why or why not?
5. How does the Mean Value Theorem apply to real-world scenarios?

Tip: Always check the continuity and differentiability of a function before applying the Mean Value Theorem.