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Chapter 3: Problem 42

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 of the form \[ f(x) = ax^2 + bx + c \]where \( a = -\frac{1}{3} \), \( b = 2 \), and \( c = 7 \). Since \( a < 0 \), the parabola opens downward, indicating a maximum point.
02

Find the Vertex

The vertex of a parabola \( ax^2 + bx + c \) occurs at \[ x = -\frac{b}{2a} \]Substitute the values of \( a \) and \( b \) from the function:\[ x = -\frac{2}{2 \times -\frac{1}{3}} = 3 \].
03

Calculate the Maximum Value

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

Vertex of a Parabola
In quadratic functions like the one given in the exercise, the vertex of the parabola is a crucial point that provides information about the function's maximum or minimum value. For a quadratic function expressed as \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). This formula gives the x-coordinate of the vertex.
  • **Finding the Vertex:** Substitute the coefficients \( a \) and \( b \) from the function into the formula to find the x-coordinate of the vertex.
  • **Example:** With \( a = -\frac{1}{3} \) and \( b = 2 \), we find \( x = -\frac{2}{2 \times -\frac{1}{3}} = 3 \).
After finding the x-coordinate of the vertex, do not forget to substitute it back into the function to determine the y-coordinate, which also represents the function's maximum or minimum value.
Maximum and Minimum Values
The maximum or minimum value of a quadratic function indicates the highest or lowest point on the parabola. Understanding this concept allows us to evaluate the function's behavior efficiently.
  • **Parabola Opening:** If \( a > 0 \), the parabola opens upward, and the vertex gives the minimum value. Conversely, if \( a < 0 \), the parabola opens downward, and the vertex signifies the maximum value.
  • **Calculating the Value:** Substitute the x-coordinate of the vertex back into the original function to find either the maximum or minimum value at that point.
  • **Example:** For our function, \( a < 0 \), so the maximum value is computed as \( f(3) = -\frac{3^2}{3} + 2 \times 3 + 7 = 10 \).
Understanding these concepts can help in identifying crucial points where the function reaches its extremities.
Parabolas
Parabolas are the graph representation of quadratic functions, showcasing distinct characteristics depending on the coefficient 'a'. They are symmetrical, and their shape can tell us a lot about the nature of the quadratic function.
  • **Symmetrical Nature:** The line of symmetry passes through the vertex, meaning the left and right sides of the parabola are mirror images of each other.
  • **Direction:** The direction in which a parabola opens is governed by the value of \( a \). If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards.
  • **Vertex Role:** The vertex serves as the peak (for \( a < 0 \)) or the lowest point (for \( a > 0 \)) of the parabola, making it vital in determining the function's key values.
From weather phenomena like the path of water from a fountain to engineering structures, the concept of a parabola and its features are incredibly useful and widely applicable.

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Most popular questions from this chapter

Find the maximum or minimum value of the function. $$f(t)=-3+80 t-20 t^{2}$$

Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ h(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

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Maximum and Minimum Values A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Sketch a graph of \(f .\) (c) Find the maximum or minimum value of \(f .\) $$f(x)=x^{2}-8 x+8$$
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