\begin{tabular}{l} Answer the questions below about the quadratic function. \\ \( \qquad g(x)=x^{2}-10 x+24 \) \\ \( \qquad \)\begin{tabular}{l} Does the function have a minimum or maximum value? \\ What is the function's minimum or maximum value? \\ \( \begin{array}{l}\text { Where does the minimum or maximum value occur? } \\ x=\square\end{array} \) \\ \hline\end{tabular} \\ \hline O Maximum \\ \hline\end{tabular}

Let's analyze the quadratic function \( g(x) = x^{2} - 10x + 24 \) to answer the questions: ### 1. Does the function have a minimum or maximum value? The coefficient of \( x^{2} \) is **positive** (\( a = 1 > 0 \)), which means the parabola opens ** upwards**. Therefore, the function **has a minimum value**. ### 2. What is the function's minimum value? To find the minimum value, we can complete the square or use the vertex formula. **Using the vertex formula:** The vertex of a parabola \( ax^{2} + bx + c \) is at \( x = -\frac{b}{2a} \). For \( g(x) = x^{2} - 10x + 24 \): - \( a = 1 \) - \( b = -10 \) \[ x = -\frac{-10}{2 \times 1} = 5 \] Now, substitute \( x = 5 \) back into the function to find the minimum value: \[ g(5) = (5)^{2} - 10(5) + 24 = 25 - 50 + 24 = -1 \] So, the **minimum value is** \(-1\). ### 3. Where does the minimum value occur? The minimum value occurs at \( x = 5 \). ### Summary: - **Type of Extremum:** Minimum - **Minimum Value:** \(-1\) - **Location:** \( x = 5 \)