Section 1
Extreme Values of Functions
Determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1.(GRAPH CANNOT COPY)
Find the absolute extreme values and where they occur.(GRAPH CANNOT COPY)
Match the table with a graph. (GRAPH CANNOT COPY)$$\begin{array}{ll}\hline x & f^{\prime}(x) \\\hline a & 0 \\b & 0 \\c & 5 \\\hline\end{array}$$
Match the table with a graph. (GRAPH CANNOT COPY)$$\begin{array}{lc}\hline x & f^{\prime}(x) \\\hline a & 0 \\b & 0 \\c & -5 \\\hline\end{array}$$
Match the table with a graph. (GRAPH CANNOT COPY)$$\begin{array}{lr}\hline x & f^{\prime}(x) \\\hline a & \text { does not exist } \\b & 0 \\c & -2 \\\hline\end{array}$$
Match the table with a graph. (GRAPH CANNOT COPY)$$\begin{array}{lc}\hline x & f^{\prime}(x) \\\hline a & \text { does not exist } \\b & \text { does not exist } \\c & -1.7 \\\hline\end{array}$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$f(x)=|x|, \quad-1<x<2$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$y=\frac{6}{x^{2}+2}, \quad-1<x<1$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$g(x)=\left\{\begin{array}{ll}-x, & 0 \leq x<1 \\x-1, & 1 \leq x \leq 2\end{array}\right.$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$h(x)=\left\{\begin{array}{l}\frac{1}{x},-1 \leq x<0 \\\sqrt{x}, 0 \leq x \leq 4\end{array}\right.$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$y=3 \sin x, \quad 0<x<2 \pi$$
Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.$$f(x)=\left\{\begin{array}{lr}x+1, & -1 \leq x<0 \\\cos x, & 0<x \leq \frac{\pi}{2}\end{array}\right.$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(x)=-x-4, \quad-4 \leq x \leq 1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(x)=x^{2}-1, \quad-1 \leq x \leq 2$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(x)=4-x^{3},-2 \leq x \leq 1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$F(x)=-\frac{1}{x},-2 \leq x \leq-1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$h(x)=\sqrt[3]{x}, \quad-1 \leq x \leq 8$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$h(x)=-3 x^{2 / 3}, \quad-1 \leq x \leq 1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=\sqrt{4-x^{2}},-2 \leq x \leq 1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=-\sqrt{5-x^{2}},-\sqrt{5} \leq x \leq 0$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=\sec x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{6}$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(t)=2-|t|, \quad-1 \leq t \leq 3$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(t)=|t-5|, \quad 4 \leq t \leq 7$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=x e^{-x}, \quad-1 \leq x \leq 1$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$h(x)=\ln (x+1), \quad 0 \leq x \leq 3$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$$
Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.$$g(x)=e^{-x^{2}}, \quad-2 \leq x \leq 1$$
Find the function's absolute maximum and minimum values and say where they are assumed.$$f(x)=x^{4 / 3}, \quad-1 \leq x \leq 8$$
Find the function's absolute maximum and minimum values and say where they are assumed.$$f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8$$
Find the function's absolute maximum and minimum values and say where they are assumed.$$g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1$$
Find the function's absolute maximum and minimum values and say where they are assumed.$$h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8$$
Determine all critical points for each function.$$y=x^{2}-6 x+7$$
Determine all critical points for each function.$$f(x)=6 x^{2}-x^{3}$$
Determine all critical points for each function.$$f(x)=x(4-x)^{3}$$
Determine all critical points for each function.$$g(x)=(x-1)^{2}(x-3)^{2}$$
Determine all critical points for each function.$$y=x^{2}+\frac{2}{x}$$
Determine all critical points for each function.$$f(x)=\frac{x^{2}}{x-2}$$
Determine all critical points for each function.$$y=x^{2}-32 \sqrt{x}$$
Determine all critical points for each function.$$g(x)=\sqrt{2 x-x^{2}}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=2 x^{2}-8 x+9$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x^{3}-2 x+4$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x^{3}+x^{2}-8 x+5$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x^{3}(x-5)^{2}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\sqrt{x^{2}-1}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x-4 \sqrt{x}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\frac{1}{\sqrt[3]{1-x^{2}}}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\sqrt{3+2 x-x^{2}}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\frac{x}{x^{2}+1}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\frac{x+1}{x^{2}+2 x+2}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=e^{x}+e^{-x}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=e^{x}-e^{-x}$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x \ln x$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=x^{2} \ln x$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\cos ^{-1}\left(x^{2}\right)$$
Find the extreme values (absolute and local) of the function over its natural domain, and where they occur.$$y=\sin ^{-1}\left(e^{x}\right)$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=x^{2 / 3}(x+2)$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=x^{2 / 3}\left(x^{2}-4\right)$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=x \sqrt{4-x^{2}}$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=x^{2} \sqrt{3-x}$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=\left\{\begin{array}{ll}4-2 x, & x \leq 1 \\x+1, & x>1\end{array}\right.$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=\left\{\begin{array}{ll}3-x, & x<0 \\3+2 x-x^{2}, & x \geq 0\end{array}\right.$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=\left\{\begin{array}{ll}-x^{2}-2 x+4, & x \leq 1 \\-x^{2}+6 x-4, & x>1\end{array}\right.$$
Find the critical points, domain endpoints, and extreme values (absolute and local) for each function.$$y=\left\{\begin{array}{ll}-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4}, & x \leq 1 \\x^{3}-6 x^{2}+8 x, & x>1\end{array}\right.$$
Give reasons for your answers.Let $f(x)=(x-2)^{2 / 3}$a. Does $f^{\prime}(2)$ exist?b. Show that the only local extreme value of $f$ occurs at $x=2$c. Does the result in part (b) contradict the Extreme Value Theorem?d. Repeat parts (a) and (b) for $f(x)=(x-a)^{2 / 3},$ replacing 2 by $a$
Give reasons for your answers.Let $f(x)=\left|x^{3}-9 x\right|$a. Does $f^{\prime}(0)$ exist?b. Does $f^{\prime}(3)$ exist?c. Does $f^{\prime}(-3)$ exist?d. Determine all extrema of $f$
A minimum with no derivative The function $f(x)=|x|$ has an absolute minimum value at $x=0$ even though $f$ is not differentiable at $x=0 .$ Is this consistent with Theorem $2 ?$ Give reasons for your answer.
If an even function $f(x)$ has a local maximum value at $x=c,$ can anything be said about the value of $f$ at $x=-c ?$ Give reasons for your answer.
If an odd function $g(x)$ has a local minimum value at $x=c,$ can anything be said about the value of $g$ at $x=-c ?$ Give reasons for your answer.
We know how to find the extreme values of a continuous function $f(x)$ by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.
The function $$V(x)=x(10-2 x)(16-2 x), \quad 0<x<5$$models the volume of a box.a. Find the extreme values of $V$b. Interpret any values found in part (a) in terms of the volume of the box.
Consider the cubic function$$f(x)=a x^{3}+b x^{2}+c x+d$$a. Show that $f$ can have $0,1,$ or 2 critical points. Give examples and graphs to support your argument.b. How many local extreme values can $f$ have?
The height of a body moving vertically is given by$$s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, \quad g>0$$with $s$ in meters and $t$ in seconds. Find the body's maximum height.
Peak alternating current Suppose that at any given time $t$ (in seconds) the current $i$ (in amperes) in an alternating current circuit is $i=2 \cos t+2 \sin t .$ What is the peak current for this circuit (largest magnitude)?
Then find the extreme values of the function on the interval and say where they occur.$$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$
Then find the extreme values of the function on the interval and say where they occur.$$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$
Then find the extreme values of the function on the interval and say where they occur.$$h(x)=|x+2|-|x-3|, \quad-\infty<x<\infty$$
Then find the extreme values of the function on the interval and say where they occur.$$k(x)=|x+1|+|x-3|, \quad-\infty<x<\infty$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=x^{2 / 3}(3-x), \quad[-2,2]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=\pi x^{2} e^{-3 x / 2}, \quad[0,5]$$
You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.a. Plot the function over the interval to see its general behavior there.b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.c. Find the interior points where $f^{\prime}$ does not exist.d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.e. Find the function's absolute extreme values on the interval and identify where they occur.$$f(x)=\ln (2 x+x \sin x), \quad[1,15]$$