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Chapter 3: Problem 38

Find the maximum or minimum value of the function. $$f(x)=6 x^{2}-24 x-100$$

Short Answer

Expert verified
The minimum value of the function is \(-124\), at \( x = 2 \).

Step by step solution

01

Identify the Form of the Function

The function given is a quadratic function, expressed in the standard form \( f(x) = ax^2 + bx + c \). For this function, \( a = 6 \), \( b = -24 \), and \( c = -100 \). A quadratic function has a parabola shape, and since \( a > 0 \), the parabola opens upwards and therefore has a minimum point.
02

Use Vertex Formula to Find Vertex

The vertex of a parabola \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substitute \( a = 6 \) and \( b = -24 \) to find the x-coordinate of the vertex: \[ x = -\frac{-24}{2 \times 6} = \frac{24}{12} = 2 \]
03

Calculate the Minimum Value at the Vertex

Now, substitute \( x = 2 \) back into the function to find the function value at this x-coordinate, which will be the minimum value: \[ f(2) = 6(2)^2 - 24(2) - 100 \] Calculate step by step: \[ f(2) = 6 \times 4 - 48 - 100 = 24 - 48 - 100 \] \[ f(2) = 24 - 48 - 100 = -124 \]
04

Conclusion

The minimum value of the function \( f(x) = 6x^2 - 24x - 100 \) is \(-124\), occurring at \( x = 2 \). This is because the parabola opens upwards and the vertex represents the lowest point on the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A quadratic function is often associated with the graph of a parabola. This is because when you plot a quadratic equation, the shape that emerges is that of a U or an upside-down U, depending on the sign of the coefficient of the quadratic term. Here are some key features of a parabola:
  • The highest or lowest point on a parabola is called the vertex.
  • If the coefficient of the quadratic term (\(a\)) is positive, the parabola opens upwards and looks like a U shape.
  • If \(a\) is negative, the parabola opens downwards, creating an upside-down U shape.
In the equation \(f(x) = 6x^2 - 24x - 100\), the positive \(a\) value indicates that the parabola is opening upwards, hence it has a minimum point rather than a maximum.
Vertex Formula
The vertex of a parabola is a crucial point because it represents either the minimum or maximum value of the function. To locate this vertex, we use the vertex formula. For a quadratic function given by \(ax^2 + bx + c\), the x-coordinate of the vertex can be calculated using the formula:\[ x = -\frac{b}{2a} \]Let's break it down further:
  • The calculation involves the coefficients \(b\) and \(a\) from the standard quadratic form.
  • In the formula \(x = -\frac{-24}{2 \times 6}\), substituting the values \(b = -24\) and \(a = 6\) gives the x-coordinate of the vertex as \(x = 2\).
Understanding this step is essential because it allows us to find where on the x-axis the vertex is located, which is critical for determining the minimum or maximum value of the quadratic function.
Minimum Value
Once the vertex is located, the next step is to find the minimum or maximum value of the function, depending on the direction of the parabola. For a parabola that opens upwards, like the one described by \(f(x) = 6x^2 - 24x - 100\), the vertex is the minimum point.Here's how to compute this minimum value:
  • With the x-coordinate of the vertex known to be \(x = 2\), substitute this back into the original equation to find \(f(2)\).
  • Calculating step by step, \(f(2) = 6(2)^2 - 24 \times 2 - 100\)
  • This simplifies to \(f(2) = 6 \times 4 - 48 - 100 = -124\).
Therefore, the function's minimum value is \(-124\), which is the lowest point on the graph of this quadratic function. This value is significant because it represents the smallest output from this function.

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