91Ó°ÊÓ

Honeycomb The surface area of a cell in a honeycomb is \(S=6 h s+\frac{3 s^{2}}{2}\left(\frac{\sqrt{3}-\cos \theta}{\sin \theta}\right)\) where \(h\) and \(s\) are positive constants and \(\theta\) is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle \(\theta(\pi / 6 \leq \theta \leq \pi / 2)\) that minimizes the surface area \(S\).

Minimize the sum of the perpendicular distances (see Exercises \(85-90\) in Section \(\mathrm{P.2}\) ) from the trunk line to the factories given by \(S_{3}=\frac{|4 m-1|}{\sqrt{m^{2}+1}}+\frac{|5 m-6|}{\sqrt{m^{2}+1}}+\frac{|10 m-3|}{\sqrt{m^{2}+1}}\) Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines. (Hint: Use a graphing utility to graph the function \(S_{3}\) and approximate the required critical number.)

Use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur? $$ f(x)=\frac{x^{3}}{x^{2}+1} $$

In Exercises 65 and 66 , use the Intermediate Value Theorem and Rolle's Theorem to prove that the equation has exactly one real solution. 65\. \(x^{5}+x^{3}+x+1=0\) 66\. \(2 x-2-\cos x=0\)Use the Intermediate Value Theorem and Rolle's Theorem to prove that the equation has exactly one real solution. $$x^{5}+x^{3}+x+1=0$$

A model for the specific gravity of water \(S\) is $$ S=\frac{5.755}{10^{8}} T^{3}-\frac{8.521}{10^{6}} T^{2}+\frac{6.540}{10^{5}} T+0.99987,0

Let the function \(f\) be differentiable on an interval \(I\) containing c. If \(f\) has a maximum value at \(x=c\), show that \(-f\) has a minimum value at \(x=c\).

Consider the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) where \(a \neq 0 .\) Show that \(f\) can have zero, one, or two critical numbers and give an example of each case.

Create a function whose graph has the given characteristics. Vertical asymptote: \(x=0\) Slant asymptote: \(y=-x\)

Sketch the graph of the arbitrary function \(f\) such that \(f^{\prime}(x)\left\\{\begin{array}{ll}>0, & x<4 \\ \text { undefined, } & x=4 \\ <0, & x>4\end{array}\right.\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of a polynomial function has three \(x\) -intercepts, then it must have at least two points at which its tangent line is horizontal.