Consider the function [tex]$f(x)=3 x^2-24 x-4$[/tex].
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
a. The function has a value.
Answer (1)
The function has a minimum value because the coefficient of the x 2 term is positive.
The x-coordinate of the vertex is found using the formula x = − b / ( 2 a ) , which gives x = 4 .
The minimum value is found by substituting x = 4 into the function, resulting in f ( 4 ) = − 52 .
The domain is ( − ∞ , ∞ ) , and the range is [ − 52 , ∞ ) . The final answer is that the function has a minimum value of − 52 .
Explanation
Problem Analysis We are given the quadratic function f ( x ) = 3 x 2 − 24 x − 4 . Our goal is to determine whether this function has a minimum or maximum value, find that value and where it occurs, and identify the function's domain and range.
Minimum or Maximum Value To determine whether the function has a minimum or maximum value, we look at the coefficient of the x 2 term. In this case, the coefficient is 3, which is positive. Since the coefficient is positive, the parabola opens upwards, and the function has a minimum value.
Finding the Vertex To find the x-coordinate of the vertex, we use the formula x = − b / ( 2 a ) , where a = 3 and b = − 24 . Thus, x = − ( − 24 ) / ( 2 ∗ 3 ) = 24/6 = 4 .
Finding the Minimum Value To find the minimum value, we substitute the x-coordinate of the vertex into the function: f ( 4 ) = 3 ( 4 ) 2 − 24 ( 4 ) − 4 = 3 ( 16 ) − 96 − 4 = 48 − 96 − 4 = − 52 . Therefore, the minimum value is -52, and it occurs at x = 4 .
Determining the Domain The domain of a quadratic function is all real numbers, which can be written as ( − ∞ , ∞ ) .
Determining the Range Since the function has a minimum value of -52, the range is [ − 52 , ∞ ) .
Final Answer The function has a minimum value of -52, which occurs at x = 4 . The domain of the function is ( − ∞ , ∞ ) , and the range is [ − 52 , ∞ ) .
Examples
Imagine you're designing a parabolic arch for a bridge. The function f ( x ) = 3 x 2 − 24 x − 4 models the shape of the arch. Finding the minimum value helps you determine the lowest point of the arch, which is crucial for ensuring sufficient clearance underneath the bridge. Understanding the domain and range ensures the arch is appropriately sized and positioned within the design constraints. This optimization ensures structural integrity and efficient use of materials.
