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Chapter 1: Problem 58

Use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. $$ f(x)=3 x^{2}-2 x-5 $$

Short Answer

Expert verified
The function \(f(x) = 3x^2 - 2x - 5\) has a relative minimum. The exact value will depend on the graph drawn and the precision of the graphing tool.

Step by step solution

01

Understand the given function

You are given the function \(f(x) = 3x^2 - 2x - 5\). This is a quadratic function. Quadratic functions have the general form \(ax^2 + bx + c\). From this, we can understand that the function opens upwards (a > 0), indicating that it will have a relative minimum but not a relative maximum.
02

Graph the function

You can graph the function using any graphing utility tool. Plot the function \(f(x) = 3x^2 - 2x - 5\) on the tool.
03

Identify the relative minimum

Once the graph is plotted, identify the lowest point on the graph. This point is known as the relative minimum of the function. The x-coordinate of this point is the value of x for which the function f has the lowest value.
04

Approximate the relative minimum value

Use the cursor, trace, or other equivalent feature on your graphing utility to approximate the coordinates of the minimum point to two decimal places. Make sure to use the y-coordinate of this point for the relative minimum value of the function.

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Most popular questions from this chapter

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

The function given by \(y=0.03 x^{2}+245.50, \quad 0

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=|x-2|, \quad x \leq 2 $$

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=5 $$

Use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ f^{-1} \circ g^{-1} $$
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