Define extremophile, and identify two “extreme” habitats.

Fill in each blank with the appropriate word or phrase.

The $\_\_\_\_\_\_\_\_\_\_\_\_\_$ is the probability, assuming $H_0$ is true, of observing a value for the test statistic that is as extreme as or more extreme than the value actually observed.

Use truth tables to determine whether or not the following arguments are valid:

(a) Either sales or expenses will go up. If sales go up, then the boss will be happy. If expenses go up, then the boss will be unhappy. Therefore, sales and expenses will not both go up.

(b) If the tax rate and the unemployment rate both go up, then there will be a recession. If the GNP goes up, then there will not be a recession. The GNP and taxes are both going up. Therefore, the unemployment rate is not going up.

(c) The warning light will come on if and only if the pressure is too high and the relief valve is clogged. The relief valve is not clogged. Therefore, the warning light will come on if and only if the pressure is too high.

Which planet has the most extreme temperature variations?

For the following problem, find the derivative function, $f^{\prime}(x)$. Use the fact that the derivative is zero at an extreme point to find the $x$-coordinates of all extreme points. Confirm your answer graphically.

$$f(x)=x^3-12 x^2+36 x+17$$

Find the extreme values of f(x, y, z) = x - y + z on the unit sphere $x^{2}+y^{2}+z^{2}=1.$

Data Analysis The table shows the unemployment rate $r$ for the years $1990$ through $2001$ in the United States. The time $t$ is measured in years, with $t=0$ corresponding to $1990.$

$$\begin{array}{|c|c|} \hline \text{Times},t & \text{Rate},r \\ \hline 0 & 5.6 \\ 1 & 6.8 \\ 2 & 7.5 \\ 3 & 6.9 \\ 4 & 6.1 \\ 5 & 5.6 \\ \hline \end{array}$$

$$\begin{array}{|r|c|} \hline \text{Times},t & \text{Rate},r \\ \hline 6 & 5.4 \\ 7 & 4.9 \\ 8 & 4.5 \\ 9 & 4.2 \\ 10 & 4.0 \\ 11 & 4.8 \\ \hline \end{array}$$

Which of the following models best represents the data? Explain your reasoning.
$(1)$ $r=1.39 \sin (0.48 t+0.42)+5.51$ $(2)$ $r=1.39 \sin (0.48 t-0.01)+5.51$ $(3)$ $r=\sin (0.10 t+5.61)+4.80$ $(4)$ $r=896 \sin (0.57 t-2.05)+6.48$

For the following problem, find the derivative function, $f^{\prime}(x)$. Use the fact that the derivative is zero at an extreme point to find the $x$-coordinates of all extreme points. Confirm your answer graphically.

$$f(x)=x^3-5 x+6$$

Why are unpredictable and extreme climate changes important?

Find the extreme values of $f(x, y)=x^{3}+y^{2}$ on the circle $x^{2}+y^{2}=1.$

What are extreme conditions? Give an example.

The data for the Great Depression (1934 and 1938) were removed because the unemployment rate was 21 % and 18 %, respectively. Do you agree that they should be removed for this investigation? Why or why not?

Can you identify two extreme habitats for extremophile organisms?

In this exercise, determine whether the variation model is of the form

$y=k x \quad$ or $\quad y=\frac{k}{x}$

and find $k$.