Prove that

$$f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 4 x$$

has no root c satisfying c > 0. Hint: Note that x = 0 is a root and apply Rolle's Theorem.

Complete the statement using <, >, or =. 6.18 lb ___ 2.88 kg

¿Quien es? A.Complete the sentences below with the correct family relationships
Mis... son los hermanos de mis padres.

Find the exact area of the region. Bounded by $y = 1 / \sqrt { x ^ { 2 } + 9 },$ y = 0, x = 0, x = 3.

Describe how Alexander Hamilton's financial program affected the economy.

Evaluate the expression. Give your answer in both radians and degrees.

$$cos^{-1}0$$

Tell whether the sum of the factors of the constant term should be positive or negative when you factor the trinomial.

$$w ^ { 2 } + 11 w + 18$$

Distinguish between the terms hyphae and mycelium.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the product of a point's coordinates is positive, the point must be in quadrant I.

In a cathode ray tube, electrons initially at rest are accelerated by a uniform electric field of magnitude $4.0 \times 10 ^ { 5 } \mathrm { N } / \mathrm { C }$ during the first 5.0 cm of the tube's length; then they move at essentially constant velocity another 45 cm before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?

Graph each equation in a

$$\left[ - 2 \pi , 2 \pi , \frac { \pi } { 2 } \right]$$

by [-3, 3, 1] viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent.

$$y = \frac { 2 \tan \frac { x } { 2 } } { 1 + \tan ^ { 2 } \frac { x } { 2 } }$$

(a) Give a counterexample to show that $$ ( A B ) ^ { - 1 } \neqA ^ { - 1 } B ^ { - 1 } $$ in general. (h) Under what conditions on A and B is $$ ( A B ) ^ { - 1 } =A ^ { - 1 } B ^ { - 1 } ? $$ Prove your assertion.

Suppose $A$ and $B$ are sets, and $f$ and $g$ are functions with $f:A\rightarrow B$ and $g:B\rightarrow A$. Prove: If $g\circ f=\text{id}_A$ and $f\circ g=\text{id}_B$, then $f$ is invertible and $g=f^{-1}$. Note: This result is a converse to Proposition 26.9.

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to

$$ℤ_p × ℤ_p$$

for some prime p.