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.