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Let's break down and solve the problem step by step based on the given graph and integral.

We are given the function:

\[
A(x) = \int_0^x f(t)\,dt
\]

with \( f(x) \) as shown in the figure. The questions are asking about the behavior of \( A(x) \), which is the integral (area under the curve) of \( f(x) \) from 0 to \( x \).

### a. Where does \( A(x) \) have its absolute maximum over the interval \([P, S]\)?
- \( A(x) \) is the accumulation of the area under the curve of \( f(x) \).
- The function \( f(x) \) appears positive in intervals before \( R \) and negative afterward. 
- The absolute maximum of \( A(x) \) is likely where the accumulation of positive area is the greatest.
  
**Answer for (a):** \( A(x) \) likely has its absolute maximum at point \( R \), where \( f(x) \) switches from positive to negative.

### b. Where does \( A(x) \) have its absolute minimum over the interval \([P, S]\)?
- The absolute minimum of \( A(x) \) occurs where the accumulated area becomes most negative.
- Since \( f(x) \) is negative after \( R \), the area under the curve decreases rapidly.

**Answer for (b):** \( A(x) \) likely has its absolute minimum at point \( S \), where the negative area is greatest.

### c. On what interval is \( A(x) \) increasing?
- \( A(x) \) increases when \( f(x) \geq 0 \), because the integral accumulates positive area.
- From the graph, \( f(x) \) is positive between points \( P \) and \( R \).

**Answer for (c):** \( A(x) \) is increasing on the interval \([P, R]\), since \( f(x) \geq 0 \) there.

Would you like further details on these answers? Here are 5 related questions for deeper understanding:

1. How is the sign of \( f(x) \) related to the increase or decrease of \( A(x) \)?
2. What is the physical interpretation of \( A(x) \) as an area function?
3. Can you explain how the Fundamental Theorem of Calculus applies to this problem?
4. How would you determine critical points of \( A(x) \) using \( f(x) \)?
5. What role does the behavior of \( f(x) \) at points \( P \), \( Q \), \( R \), and \( S \) play in determining the extrema of \( A(x) \)?

**Tip:** Understanding the relation between the graph of a function and its integral can greatly help with analyzing the function’s accumulated area behavior.

Let A(x) = ∫ f(t) dt, with f(x) as in Figure 12.
Where does A have its absolute maximum over the interval [P, S]?
Where does A have its absolute minimum over the interval [P, S]?
On what interval is A increasing?

This problem focuses on analyzing the behavior of A(x) = ∫ f(t) dt, where f(x) is represented by a graph. Learn to determine the absolute maximum and minimum of A(x) over the interval [P, S], as well as the interval where A(x) is increasing. Key concepts include integrals, area under the curve, and the Fundamental Theorem of Calculus. Suitable for advanced high school students (Grades 11-12).

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