f. What is the smallest interval \( [a, b] \) on which values of the Gini index lie for \( L(x)=x^{p} \) with \( p \geq 1 \) ? The minimum value of \( G \) is \( \square \) and the maximum value of \( G \) is \( \square \).

To determine the smallest interval \([a, b]\) on which the Gini index \(G\) lies for the Lorenz curve \(L(x) = x^p\) with \(p \geq 1\), we can follow these steps: ### **1. Understanding the Gini Index** The Gini index is a measure of inequality within a distribution, often used to represent income or wealth distribution. Mathematically, it's defined based on the Lorenz curve \(L(x)\) as: \[ G = 1 - 2 \int_0^1 L(x) \, dx \] ### **2. Calculating the Gini Index for \(L(x) = x^p\)** Given \(L(x) = x^p\), where \(p \geq 1\): \[ \int_0^1 L(x) \, dx = \int_0^1 x^p \, dx = \left[ \frac{x^{p+1}}{p+1} \right]_0^1 = \frac{1}{p+1} \] Substituting this into the Gini index formula: \[ G = 1 - 2 \left( \frac{1}{p+1} \right) = 1 - \frac{2}{p+1} \] ### **3. Determining the Range of \(G\)** - **Minimum Value of \(G\):** When \(p = 1\): \[ G = 1 - \frac{2}{1+1} = 1 - 1 = 0 \] This represents perfect equality. - **Maximum Value of \(G\):** As \(p \to \infty\): \[ G = 1 - \frac{2}{p+1} \to 1 \] This indicates extreme inequality. ### **4. Conclusion** The Gini index \(G\) for the given Lorenz curve \(L(x) = x^p\) with \(p \geq 1\) lies within the interval \([0, 1]\). - **Minimum value of \(G\):** \(0\) - **Maximum value of \(G\):** \(1\) --- **Answer:** The Gini index \(G\) lies in the interval \([0,\,1]\). Its minimum value is 0 and its maximum value is 1.