91Ó°ÊÓ

Determine the open intervals on which the graph is concave upward or concave downward. $$ y=x^{2}-x-2 $$

Find two positive numbers whose sum is 110 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) \begin{tabular}{|c|c|c|} \hline First Number \(x\) & Second Number & Product \(P\) \\ \hline 10 & \(110-10\) & \(10(110-10)-1000\) \\ \hline 20 & \(110-20\) & \(20(110-20)=1800\) \\ \hline \end{tabular} (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (Hint: Use the table feature of the graphing utility.) (c) Write the product \(P\) as a function of \(x\). (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use calculus to find the critical number of the function in part (c). Then find the two numbers.

Graph the fourth-degree polynomial \(p(x)=x^{4}+a x^{2}+1\) for various values of the constant \(a .\) (a) Determine the values of \(a\) for which \(p\) has exactly one relative minimum. (b) Determine the values of \(a\) for which \(p\) has exactly one relative maximum. (c) Determine the values of \(a\) for which \(p\) has exactly two relative minima. (d) Show that the graph of \(p\) cannot have exactly two relative extrema.

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. \begin{tabular}{|c|c|c|} \hline Height & Length and Width & Volume \\ \hline 1 & \(24-2(1)\) & \(1[24-2(1)]^{2}=484\) \\ \hline 2 & \(24-2(2)\) & \(2[24-2(2)]^{2}=800\) \\ \hline \end{tabular} (b) Write the volume \(V\) as a function of \(x\). (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Prove Darboux's Theorem: Let \(f\) be differentiable on the closed interval \([a, b]\) such that \(f^{\prime}(a)=y_{1}\) and \(f^{\prime}(b)=y_{2} .\) If \(d\) lies between \(y_{1}\) and \(y_{2}\), then there exists \(c\) in \((a, b)\) such that \(f^{\prime}(c)=d\)

Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=\frac{2 x}{x^{2}-1} $$

Numerical and Graphical Analysis. use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ f(x)=\frac{8 x}{\sqrt{x^{2}-3}} $$

In an autocatalytic chemical reaction, the product formed is a catalyst for the reaction. If \(Q_{0}\) is the amount of the original substance and \(x\) is the amount of catalyst formed, the rate of chemical reaction is \(\frac{d Q}{d x}=k x\left(Q_{0}-x\right)\) For what value of \(x\) will the rate of chemical reaction be greatest?

Apply Newton's Method to approximate the \(x\) -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. [Hint: Let \(h(x)=f(x)-g(x) .]\) $$ \begin{aligned} &f(x)=x^{2} \\ &g(x)=\cos x \end{aligned} $$

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?