Let’s analyze and solve the given problem step by step.

Problem Breakdown:

1. Cumulative area function: The cumulative area function $F(x)$ is defined as: $F(x) = \int_1^x f(t) \, dt$

   • $F(1) = 0$ because the integral starts and ends at the same point.
   • The sign of the area depends on whether $f(x)$ is above (positive) or below (negative) the x-axis.

2. Areas given:

   • Region $A$ (from $x = 1$ to $x = 2$) has an area of $6$ (positive since $f(x) > 0$).
   • Region $B$ (from $x = 2$ to $x = 3$) has an area of $6$ (positive since $f(x) > 0$).
   • Region $C$ (from $x = 3$ to $x = 4$) has an area of $4$ (negative since $f(x) < 0$).

(a) Calculating $F(x)$ at the given points:

• $F(1) = 0$
• $F(2) = F(1) + \text{area of region A} = 0 + 6 = 6$
• $F(3) = F(2) + \text{area of region B} = 6 + 6 = 12$
• $F(4) = F(3) + \text{area of region C} = 12 - 4 = 8$

$F(1) = 0, \quad F(2) = 6, \quad F(3) = 12, \quad F(4) = 8$

(b) Where $F(x)$ is increasing or decreasing:

• $F(x)$ is increasing where $f(x) > 0$, i.e., on $(1, 3)$.
• $F(x)$ is decreasing where $f(x) < 0$, i.e., on $(3, 4)$.

$F(x) \text{ is increasing on } (1, 3), \quad F(x) \text{ is decreasing on } (3, 4).$

(c) Where $F(x)$ is concave up or down:

• $F(x)$ is concave up where $f'(x) = f(x) > 0$ is increasing, i.e., on $(1, 2)$.
• $F(x)$ is concave down where $f'(x) = f(x) > 0$ is decreasing, i.e., on $(2, 3)$.

$F(x) \text{ is concave up on } (1, 2), \quad F(x) \text{ is concave down on } (2, 3).$

(d) Sketching $F(x)$:

To sketch:

1. Plot points: $F(1) = 0$, $F(2) = 6$, $F(3) = 12$, $F(4) = 8$.
2. Draw $F(x)$ increasing from $(1, 3)$, concave up on $(1, 2)$, and concave down on $(2, 3)$.
3. Decrease $F(x)$ from $(3, 4)$ since $f(x) < 0$.

Would you like me to create a visual sketch of $F(x)$?

Follow-up Questions:

1. Can you derive $F'(x)$ and interpret it in terms of $f(x)$?
2. How does concavity change the shape of $F(x)$?
3. What happens to $F(x)$ as $f(x)$ approaches zero?
4. Can $F(x)$ ever decrease if $f(x) \geq 0$?
5. How would the graph change if the sign of the areas (A, B, C) flipped?

Tip:

When sketching $F(x)$, always focus on intervals where $f(x) > 0$ and $f(x) < 0$ to determine whether the cumulative area function is increasing or decreasing.