91Ó°ÊÓ

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) $$f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi$$

a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v\) can be modeled by the equation $$v=c\left(r_{0}-r\right) r^{2} \mathrm{cm} / \mathrm{sec}, \quad \frac{r_{0}}{2} \leq r \leq r_{0},$$ where \(r_{0}\) is the rest radius of the trachea in centimeters and \(c\) is a positive constant whose value depends in part on the length of the trachea. Show that \(v\) is greatest when \(r=(2 / 3) r_{0} ;\) that is, when the trachea is about \(33 \%\) contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. b. Take \(r_{0}\) to be 0.5 and \(c\) to be 1 and graph \(v\) over the interval \(0 \leq r \leq 0.5 .\) Compare what you see with the claim that \(v\) is at a maximum when \(r=(2 / 3) r_{0}\).

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 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. $$y=\left\\{\begin{array}{ll}-x^{2}-2 x+4, & x \leq 1 \\\\-x^{2}+6 x-4, & x>1\end{array}\right.$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll}-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4}, & x \leq 1 \\\x^{3}-6 x^{2}+8 x, & x>1\end{array}\right.$$

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.

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?

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)?

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. $$f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]$$