Let f be a constant function-that is, let f(x)=c, where c is some real number. Show that every number a gives rise to an absolute maximum and, at the same time, an absolute minimum of f.

Universal Instruments found that the monthly demand for its new line of Galaxy Home Computers r months after placing the line on the market was given by $D(t)=2000-1500 e^{-0.05 t}$ (t>0) Graph this function and answer the following question: What is the demand after 1 month? After 1 year? After 2 years? After 5 years?

For what value of c is the vector $\mathbf{v}=\langle 2,-5, c\rangle$ orthogonal to $\mathbf{w}=\langle 3,2,9\rangle$?

In each of parts (a) to (e) give the number of nonisomorphic abelian groups of the specified order-do not list the groups: (a) order 100, (b) order 576, (c) order 1155,(d) order 42875, (e) order 2704.

Use elementary row operations to transform each augmented coefficient matrix to echelon form. Then solve the system by back substitution.

$$\begin{array}{l}2 x_{1}+5 x_{2}+12 x_{3}=6 \\ 3 x_{1}+x_{2}+5 x_{3}=12 \\ 5 x_{1}+8 x_{2}+21 x_{3}=17\end{array}$$

Lysozyme is an enzyme that cleaves cell walls. A 0.100-L sample of a solution of lysozyme that contains 0.0750 g of the enzyme exhibits an osmotic pressure of 1.32 $\times 10^{-3}$ atm at 25 $^{\circ} \mathrm{C}$. What is the molar mass of lysozyme?

An electrical load has a power factor of 0.8 lagging. It dissipates 8 kW at a voltage of 200 V. Calculate the impedance of this load in rectangular coordinates.

The following problem use the method of elimination to determine whether the given linear system is consistent or inconsistent. For each consistent system, find the solution if it is unique; otherwise, describe the infinite solution set in terms of an arbitrary parameter t. x + 3y + 2z = 2, 2x + 7y + 7z = -1, 2x + 5y + 2z = 7

Prove that every non-abelian group of order 6 has a nonnormal subgroup of order 2. Use this to classify groups of order 6. [Produce an injective homomorphism into $S_{3}$.]

The number of banks in the United States has been dropping steadily since 1984, and the trend in recent years has been roughly linear. The annual data for the years 2003 through 2012 can be summarized as follows, where x represents the years since 2000 and y the number of banks, in thousands, in the United States.

$$\begin{array}{ccc} n=10 & \Sigma x=75 & \Sigma x^{2}=645 \\ \Sigma y=70.457 & \Sigma y^{2}=499.481335 & \Sigma x y=512.775 \end{array}$$

(a) Find an equation for the least squares line. (b) Use your result from part (a) to predict the number of U.S. banks in the year 2020. (c) If this trend continues linearly, in what year will the number of U.S. banks drop below 4000? (d) Find and interpret the correlation coefficient.

(a) state the null and alternative hypotheses and identify which represents the claim, (b) describe type I and type II errors for a hypothesis test of the claim, (c) explain whether the hypothesis test is left-tailed, right-tailed, or two-tailed, (d) explain how you should interpret a decision that rejects the null hypothesis, and (e) explain how you should interpret a decision that fails to reject the null hypothesis. A nonprofit consumer organization says that the standard deviation of the fuel economies of its top-rated vehicles for a recent year is no more than 9.5 miles per gallon.

In an article on the sociology of conflict, the author proposes a game in which a member of a race (she calls them the green race) visits a country in which some natives are progreen, some indifferent, and some antigreen. The green visitor, unable to determine which group a native is from, has three options: act appeasing, expect civility but not fight for it, and expect civility and fight for it if he doesn't get it. The author suggests the following payoff matrix.

$$\begin{matrix} \text{ } & \text{Pro-green} & \text{Indifferent} & \text{Anti-green}\\ \text{Appease} & \text{0} & \text{0} & \text{-4}\\ \text{No Fight} & \text{0} & \text{-3} & \text{-1}\\ \text{Fight} & \text{-2} & \text{-1} & \text{-1}\\ \end{matrix}$$

What is the visitor's optimum strategy, and what is the value of the game?

Let $Z_{n}$ be a cyclic group of order $n$ and for each integer a let

$$\sigma_{a}: Z_{n} \rightarrow Z_{n} \quad \text { by } \quad \sigma_{a}(x)=x^{a} \text { for all } x \in Z_{n}$$

(a) Prove that $\sigma_{a}$ is an automorphism of $Z_{n}$ if and only if a and n are relatively prime. (b) Prove that $\sigma_{a}=\sigma_{b}$ if and only if $a \equiv b(\text{mod} n)$. (c) Prove that every automorphism of $Z_{n}$ is equal to $\sigma_{a}$ for some integer a. (d) Prove that $\sigma_{a} \circ \sigma_{b}=\sigma_{a b}$. Deduce that the map $\bar{a} \mapsto \sigma_{a}$ is an isomorphism of $(\mathbb{Z} / n \mathbb{Z})^{\times}$ onto the automorphism group of $Z_{n}$ (so Aut $\left(Z_{n}\right)$ is an abelian group of order $\varphi(n)$).

In general, people tend to live longer in countries that have a greater supply of food. Listed in the table at the top of the next column is the 2009 daily calorie supply and 2009 life expectancy at birth for 10 randomly selected countries.

$$\begin{array}{|lcc|} \hline \text { Country } & \text { Calories }(x) & \text { Life Expectancy }(y) \\ \hline \text { Belize } & 2680 & 75.6 \\ \hline \text { Cambodia } & 2382 & 62.7 \\ \hline \text { France } & 3531 & 81.2 \\ \hline \text { India } & 2321 & 64.7 \\ \hline \text { Mexico } & 3146 & 76.5 \\ \hline \text { New Zealand } & 3172 & 80.4 \\ \hline \text { Peru } & 2563 & 73.6 \\ \hline \text { Sweden } & 3125 & 81.1 \\ \hline \text { Tanzania } & 2137 & 56.6 \\ \hline \text { United States } & 3668 & 78.2 \\ \hline \end{array}$$

(a) Find the correlation coefficient. Do the data seem to fit a straight line? (b) Draw a scatterplot of the data. Combining this with your results from part (a), do the data seem to fit a straight line? (c) Find the equation of the least squares line. (d) Use your answer from part (c) to predict the life expectancy in Canada, which has a daily calorie supply of 3399. Compare your answer with the actual value of 80.8 years. (e) Briefly explain why countries with a higher daily calorie supply might tend to have a longer life expectancy. Is this trend likely to continue to higher calorie levels? Do you think that an American who eats 5000 calories a day is likely to live longer than one who eats 3600 calorie? Why or why not? (f) Find the correlation coefficient and least squares line using the data for a larger sample of countries, as found in an almanac or other reference. Is the result in general agreement with the previous results?