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Chapter 3: Problem 40

Find the maximum or minimum value of the function. $$f(x)=-\frac{x^{2}}{3}+2 x+7$$

Short Answer

Expert verified
The maximum value of the function is 10.

Step by step solution

01

Identify the Type of Function

The given function is a quadratic function, \(f(x) = -\frac{x^2}{3} + 2x + 7\), which is in the standard form \(ax^2 + bx + c\). Here, \(a = -\frac{1}{3}\), \(b = 2\), and \(c = 7\). Since \(a < 0\), the parabola opens downwards, indicating the presence of a maximum point.
02

Find the Vertex

To find the maximum or minimum value, we need to find the vertex of the parabola. The x-coordinate of the vertex of a quadratic function \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substitute \(b = 2\) and \(a = -\frac{1}{3}\) into the formula:\[x = -\frac{2}{2(-\frac{1}{3})} = -\frac{2}{-\frac{2}{3}} = 3\\]
03

Calculate the Maximum Value

Substitute \(x = 3\) back into the function \(f(x) = -\frac{x^2}{3} + 2x + 7\) to find the maximum value:\[f(3) = -\frac{3^2}{3} + 2(3) + 7 = -\frac{9}{3} + 6 + 7 = -3 + 6 + 7 = 10\\]
04

Conclude the Results

The vertex represents the maximum point of the parabola, and thus \(f(x)\) reaches its maximum value when \(x = 3\). The maximum value of the function is \(10\).

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Key Concepts

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

Parabola
A parabola is a specific, symmetrical curve that is shaped like an arch. Parabolas come in two varieties: those that open upwards and those that open downwards.
For any quadratic function of the form \( ax^2 + bx + c \), the parabola will open upwards if \( a > 0 \) and downwards if \( a < 0 \). This is crucial because it tells us whether the vertex represents a minimum or maximum point.
In our original function \( f(x) = -\frac{x^2}{3} + 2x + 7 \), the value of \( a \) is \(-\frac{1}{3}\), which is less than 0. Therefore, we know the parabola opens downwards and the vertex is the maximum point.
Vertex Formula
The vertex formula is a critical tool for quickly finding the vertex of a parabola described by a quadratic function. The x-coordinate of the vertex for the equation \( ax^2 + bx + c \) is found using: \[ x = -\frac{b}{2a} \]
In our example, substituting \( a = -\frac{1}{3} \) and \( b = 2 \) into the formula gives \( x = 3 \).
The vertex represents the highest point on a downward-opening parabola or the lowest on an upward-opening one. Knowing how to find it allows you to determine this key feature of a quadratic graph.
Maximum Value
The maximum value of a quadratic function occurs at the vertex when the parabola opens downwards. Once you've found the x-coordinate of the vertex using the vertex formula, you substitute it back into the original function to find the function's value at that point.
For our function \( f(x) = -\frac{x^2}{3} + 2x + 7 \), we found that \( x = 3 \). Substituting \( x = 3 \) into the function gives us:
  • \( f(3) = -\frac{3^2}{3} + 2(3) + 7 = 10 \)

Thus, the maximum value of the function occurs at \( x = 3 \) and is equal to 10.
Quadratic Equation
Quadratic equations are polynomial equations of degree 2, usually in the form \( ax^2 + bx + c = 0 \). However, when discussing quadratic functions like \( f(x) = ax^2 + bx + c \), these equations define the shape and position of a parabola on the coordinate plane.
Quadratic functions can describe various phenomena in fields like physics, engineering, and economics due to their predictability and simplicity. Solving or evaluating them often involves:
  • Identifying whether they have maximum or minimum values
  • Finding roots or intercepts
  • Determining key characteristics such as vertex and axis of symmetry
Understanding these elements is fundamental to mastering quadratic equations and their related functions.

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Most popular questions from this chapter

Give an example of a rational function that has vertical asymptote \(x=3 .\) Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept 4.

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{2 x^{2}+6 x+6}{x+3}, g(x)=2 x$$

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{-3+5 i}{15 i}$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{1-\sqrt{-1}}{1+\sqrt{-1}}$$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40]$$
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