Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value, \( f(x)=3 x^{2}+18 x-5 \) Does the quadratic function f have a minimum value or a maximum value? \( x \) The function \( f \) has a maximum value. What is this minimum or maximum value? (Simplify your answer.)

To determine whether the quadratic function \( f(x) = 3x^2 + 18x - 5 \) has a maximum or minimum value, we can analyze the coefficient of the \( x^2 \) term. 1. **Identify the coefficient of \( x^2 \)**: - The coefficient of \( x^2 \) is \( a = 3 \). 2. **Determine if it is a maximum or minimum**: - If \( a > 0 \), the parabola opens upwards, and the function has a minimum value. - If \( a < 0 \), the parabola opens downwards, and the function has a maximum value. - Since \( a = 3 > 0 \), the function \( f \) has a minimum value. 3. **Find the minimum value**: - The minimum value of a quadratic function can be found using the vertex formula \( x = -\frac{b}{2a} \), where \( b \) is the coefficient of \( x \). - Here, \( b = 18 \) and \( a = 3 \). Now, let's calculate the vertex \( x \): \[ x = -\frac{b}{2a} = -\frac{18}{2 \cdot 3} = -\frac{18}{6} = -3 \] 4. **Substitute \( x = -3 \) back into the function to find the minimum value**: \[ f(-3) = 3(-3)^2 + 18(-3) - 5 \] Calculating this step by step: \[ f(-3) = 3(9) + 18(-3) - 5 \] \[ = 27 - 54 - 5 \] \[ = 27 - 54 - 5 = -32 \] Thus, the minimum value of the function \( f(x) \) is \( -32 \). ### Summary: - The function \( f \) has a **minimum value**. - The minimum value is \( -32 \).