f. What is the smallest interval $$ [a, b] $$ on which values of the $$ G $$ ini index lie for $$ L(x)=x^{p} $$ with $$ p \geq 1 $$ ? The minimum value of $$ G $$ is 0 and the maximum value of $$ G $$ is 1 . Which endpoints of $$ [a, b] $$ correspond to the least and most equitable distribution of wealth? $$ G=\square $$ corresponds to the least equitable and $$ G=\square $$ corresponds to the most equitable distribution of wealth.

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The **Gini index** measures the inequality of a distribution, such as income or wealth distribution, and it ranges between 0 and 1:

- **0** represents perfect equality, where everyone has the same wealth.
- **1** represents maximal inequality, where one individual has all the wealth, and everyone else has none.

Given the function $$ L(x) = x^{p} $$ with $$ p \geq 1 $$, the smallest interval $$[a, b]$$ on which the Gini index $$ G $$ lies is:

$$
[0, 1]
$$

Within this interval:

- **$$ G = 1 $$** corresponds to the **least equitable distribution** of wealth. This means that wealth is concentrated in the hands of a single individual or a very small group, with everyone else having little to none.
  
- **$$ G = 0 $$** corresponds to the **most equitable distribution** of wealth. In this scenario, wealth is perfectly evenly distributed among all individuals, with no disparities whatsoever.

So, the filled-in statements are:

$$
G = 1 \text{ corresponds to the least equitable distribution of wealth and } G = 0 \text{ corresponds to the most equitable distribution of wealth.}
$$

**Summary:**

- **Smallest Interval for $$ G $$:** $$[0, 1]$$
- **Least Equitable Distribution:** $$ G = 1 $$
- **Most Equitable Distribution:** $$ G = 0 $$

**Final Statement:**

The smallest interval is [0, 1]. G = 1 corresponds to the least equitable distribution of wealth and G = 0 corresponds to the most equitable distribution of wealth.
The smallest interval for the Gini index $$ G $$ is $$[0, 1]$$. - $$ G = 1 $$ represents the least equitable distribution of wealth. - $$ G = 0 $$ represents the most equitable distribution of wealth. So, the endpoints are: - $$ G = 1 $$ for the least equitable distribution. - $$ G = 0 $$ for the most equitable distribution.

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