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The image contains a mathematical problem related to the rate of change of the water volume in a reservoir. Here's the breakdown of the problem:

### Problem Description:
The rate of change of water volume in a reservoir can be approximated by the function:

\[
g(t) = \frac{1}{4}t^3 - \frac{11}{4}t^2 - 4t + 44
\]

where \( t \) represents the number of months since the observation began, and \( g(t) \) is measured in units of \( 10,000 \, m^3 \) per month.

#### Part (a):
- **Question**: When does the rate of change reach its lowest value? Describe the significance of this moment for the function. In which time periods does the water volume in the reservoir increase? At what time does the water volume increase most strongly?

#### Part (b):
- **Initial Condition**: At the beginning of the observation, there were 1.8 million cubic meters of water in the reservoir. Sketch a possible graph of the function.

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### Steps to Solve:

#### Part (a):
1. **Find the lowest point (extremum) of the rate of change function \( g(t) \)**:
   - We need to compute the derivative \( g'(t) \), set it equal to zero, and solve for \( t \). This gives us the critical points.
   - After finding the critical points, we can use the second derivative test to determine if the value at these points represents a minimum.

2. **Determine when the water volume increases**:
   - The water volume increases when \( g(t) > 0 \), meaning the rate of change is positive.

3. **Find when the water volume increases the most**:
   - The strongest increase occurs at the maximum value of \( g(t) \), which we will find by analyzing the derivative.

#### Part (b):
1. **Graph Sketch**:
   - Based on the critical points and behavior of the function, we can sketch the graph, starting with an initial value of 1.8 million \( m^3 \).

I will now compute the derivative to find the required points for part (a).

The rate of change of the water volume in a reservoir is described by the function g(t) = (1/4)t^3 - (11/4)t^2 - 4t + 44, where t is the number of months since the observation began, and g(t) is in 10,000 m³ per month. When does the rate of change reach its lowest value, and during which time periods does the water volume increase the most?

Solve an optimization problem where the rate of change of water volume in a reservoir is described by a cubic function. This problem involves finding critical points, determining when the rate of change is minimal, and identifying when the water volume increases most rapidly.

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