/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Find the critical numbers of \(f... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 31

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=5-|x-5| $$

Short Answer

Expert verified
The function \(f(x)=5-|x-5|\) has a critical point at \(x = 5\). It increases on the interval \(x < 5\), and decreases on the interval \(x > 5\). The function also has a local maximum at \(x = 5\). This is confirmed by graphing the function.

Step by step solution

01

Finding the Critical Points

The critical points of a function occur where the derivative of the function is zero or undefined. However, for functions involving absolute values, we must consider the changes in direction separately. In this case, the absolute function \(|x-5|\) changes direction at \(x = 5\). Thus, \(x=5\) is a critical point.
02

Determining the Intervals

The function's behavior changes at critical points. Therefore, we need to test the nature of function \(f\) for \(x < 5\) and \(x > 5\). If \(x < 5\), \(f(x) = 5 - (5 - x) = x\). If \(x > 5\), \(f(x) = 5 - (x - 5) = 10 - x\). Thus, the function is increasing for \(x < 5\) and decreasing for \(x > 5\).
03

Determining the Extrema

A local maximum occurs when the function changes from increasing to decreasing. Since our function increases for \(x < 5\) and decreases for \(x > 5\), there is a local maximum at \(x = 5\). By plugging \(x = 5\) into \(f(x)\), we find that the maximum value is \(f(5) = 5\).
04

Verifying the Results Graphically

By plotting the function graphically, we can verify that \(x = 5\) is a local maximum and the function increases and decreases where we have determined.

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Most popular questions from this chapter

In Exercises 61 and 62, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f, P_{1},\) and \(P_{2}\) and their first derivatives at \(x=a .\) How do the approximations change as you move farther away from \(x=a\) ? \(\begin{array}{ll}\text { Function } & \frac{\text { Value of } a}{a} \\\ f(x)=2(\sin x+\cos x) & a=\frac{\pi}{4}\end{array}\)

(a) Let \(f(x)=x^{2}\) and \(g(x)=-x^{3}+x^{2}+3 x+2 .\) Then \(f(-1)=g(-1)\) and \(f(2)=g(2) .\) Show that there is at least one value \(c\) in the interval (-1,2) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c)) .\) Identify \(c .\) (b) Let \(f\) and \(g\) be differentiable functions on \([a, b]\) where \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one value \(c\) in the interval \((a, b)\) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c))\).

The function \(f\) is differentiable on the interval [-1,1] . The table shows the values of \(f^{\prime}\) for selected values of \(x\). Sketch the graph of \(f\), approximate the critical numbers, and identify the relative extrema. $$\begin{array}{|l|c|c|c|c|} \hline x & -1 & -0.75 & -0.50 & -0.25 \\ \hline f^{\prime}(x) & -10 & -3.2 & -0.5 & 0.8 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0 & 0.25 & 0.50 & 0.75 & 1 \\ \hline \boldsymbol{f}^{\prime}(\boldsymbol{x}) & 5.6 & 3.6 & -0.2 & -6.7 & -20.1 \\ \hline \end{array}$$

Average Cost A business has a cost of \(C=0.5 x+500\) for producing \(x\) units. The average cost per unit is \(\bar{C}=\frac{C}{x},\) Find the limit of \(\bar{C}\) as \(x\) approaches infinity.

(a) Graph \(f(x)=\sqrt[3]{x}\) and identify the inflection point. (b) Does \(f^{\prime \prime}(x)\) exist at the inflection point? Explain.
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