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.

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.