The following graph shows a rough approximation of historical and projected median home prices for a country for the period 2000-2024. Here, t is time in years since the start of 2000, and C(t) is the median home price in thousands of dollars. The locations of stationary points and points of inflection are indicated on the graph. Analyze the graphs important features, and interpret each feature in terms of the median home price. (a) Fill in the blanks. 
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
- **Y-axis (Vertical):** Median Home Price (in thousands of dollars)
- **X-axis (Horizontal):** Years since the start of 2000 (t)
The graph illustrates the median home price, C(t), over 24 years. Specific points, such as stationary points and points of inflection, are indicated.
### Key Features:
1. **Median Home Price in 2000:**
- Starting point (t = 0)
- \( C(0) \) = **___** thousand dollars.
2. **Low Points:**
- First low: Year t = **___**, Price = **___** thousand dollars.
- Second low: Year t = **___**, Price = **___** thousand dollars.
3. **Peaks:**
- Year t = **___**: Maximum Price = **___** thousand dollars.
4. **Trends:**
- Decreasing most rapidly: Start of year **___**, Price = **___** thousand dollars.
- Increasing most rapidly: Start of year **___**, Price = **___** thousand dollars.
### Long-Term Projection:
Assuming the trends continue indefinitely:
- Median home price is expected to approach **___** thousand dollars in the long term.
This analysis provides insight into the fluctuations and overall trajectory of median home prices in this time period, offering a helpful reference for economic and real estate studies.
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