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Chapter 3: Problem 80

Find the value of \(b\) or \(c\) that gives the function the given minimum or maximum value. $$f(x)=-x^{2}+b x-2 ; \text { maximum value } 7$$

Short Answer

Expert verified
The values of b can be 6 or -6.

Step by step solution

01

Identify the type of quadratic function

The given function is in the form of a quadratic equation: \[f(x) = -x^2 + bx - 2\].Since the coefficient of \(x^2\) is negative, this is a parabola that opens downwards and thus has a maximum value.
02

Determine the vertex

The vertex form of a quadratic function is given by: \[f(x) = a(x - h)^2 + k\],where \((h, k)\) is the vertex of the parabola.For a quadratic equation in the form \(ax^2 + bx + c\), the vertex \(x\)-coordinate \(h\) is found using the formula: \[h = -\frac{b}{2a}\].Here, \(a = -1\), so the vertex is at: \[x = -\frac{b}{2(-1)} = \frac{b}{2}\].
03

Calculate the maximum value

Given that the maximum value is 7, set the function equal to 7 when \(x = \frac{b}{2}\):\[f\left(\frac{b}{2}\right) = -\left(\frac{b}{2}\right)^2 + b\left(\frac{b}{2}\right) - 2 = 7\].
04

Simplify and solve for b

Expand and simplify the equation:\[-\left(\frac{b}{2}\right)^2 + b\left(\frac{b}{2}\right) - 2 = 7\]\[-\frac{b^2}{4} + \frac{b^2}{2} - 2 = 7\]Combine the terms involving \(b\):\[\frac{b^2}{2} - \frac{b^2}{4} - 2 = 7\]\[\frac{2b^2}{4} - \frac{b^2}{4} - 2 = 7\]\[\frac{b^2}{4} - 2 = 7\]Isolate \(b^2\):\[\frac{b^2}{4} = 9\]\[b^2 = 36\]Solve for \(b\):\[b = 6\] or \[b = -6\].

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Key Concepts

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

Quadratic Equations
Quadratic equations are mathematical expressions of the form \[ ax^2 + bx + c = 0 \]. These equations graph as parabolas in the coordinate system. They can have either a maximum or a minimum value, depending on the coefficient of the term \(x^2\).
  • If the coefficient of \(x^2\) is positive, the parabola opens upward and has a minimum value.
  • If the coefficient is negative, it opens downward and has a maximum value.
In our exercise, the quadratic function is \[-x^2 + bx - 2\]. Here, the coefficient of \(x^2\) is negative, indicating that the parabola opens downwards, meaning it will have a maximum value.
Vertex Form
To analyze quadratic functions, we often convert them to the vertex form, which is \[ f(x) = a(x - h)^2 + k \]. Here,
  • \(a\) determines the parabola's direction and width (whether it opens up or down).
  • \(h\) is the x-coordinate of the vertex.
  • \(k\) is the y-coordinate of the vertex.
For the given quadratic function \( f(x) = -x^2 + bx - 2 \), we calculate the vertex.

The x-coordinate of the vertex for a quadratic equation \(ax^2 + bx + c\) is found using the formula \[ h = -\frac{b}{2a} \]. In our equation, \(a = -1\) and \(b\) is unknown, so the vertex's x-coordinate is \( h = \frac{b}{2}\) after simplifying.
To find the maximum value, we substitute \( x = \frac{b}{2} \) back into the function and set the equation to the given maximum value.
Maximum Value
The concept of a maximum value in quadratic functions is crucial. When a parabola opens downwards, it reaches the highest possible y-value at the vertex. In our problem, we know that the function has a maximum value of 7. To find the specific value of \(b\) that gives this maximum, we substitute \( x = \frac{b}{2} \) in the quadratic equation and solve for \(b\).

The critical steps include:
  • Substitute \( x = \frac{b}{2} \) in \( f(x) = -x^2 + bx - 2 \).
  • Set \(f\left(\frac{b}{2}\right) = 7\):
  • Simplify and solve the equation:
  • From \[-\left(\frac{b}{2}\right)^2 + b\left(\frac{b}{2}\right) - 2 = 7\], expand and combine like terms.
  • After simplification, we get \( \frac{b^2}{4} = 9 \).
  • Solving for \(b\), we find \( b = 6\) or \( b = -6 \).
Thus, the values of \(b\) that make the function reach a maximum value of 7 are 6 or -6.

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Most popular questions from this chapter

A rectangular quilt is to be made so that the length is 1.2 times the width. The quilt must be between \(72 \mathrm{ft}^{2}\) and \(96 \mathrm{ft}^{2}\) to cover the bed. Determine the restrictions on the width so that the dimensions of the quilt will meet the required area. Give exact values and the approximated values to the nearest tenth of a foot.

Explain why the fundamental theorem of algebra does not apply to \(f(x)=\sqrt{x}+3\). That is, no complex number \(c\) exists such that \(f(c)=0\)

Find all fourth roots of 1 , by solving the equation \(x^{4}=1\).

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{4-x}-6 \geq 0 $$

Explain why a polynomial with real coefficients of degree 3 must have at least one real zero.
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