answer:Given that the graph of the function f(x)=cos (2x+φ) is symmetric about the point (frac{4π}{3},0), we have: 2times frac{4π}{3}+φ =kπ+ frac{π}{2}quad kin mathbb{Z} Given - frac {π}{2} < φ < frac {π}{2}, we can solve for φ and obtain φ =- frac {π}{6}. Therefore, the function f(x)=cos (2x- frac {π}{6}). Now, let's find the expression for y=f(x+ frac {π}{3}): y=f(x+ frac {π}{3})=cos [2(x+ frac {π}{3})- frac {π}{6}]=cos (2x+ frac {π}{2})=-sin 2x We observe that y=f(x+frac{π}{3}) is an odd function since f(-x)=-f(x). To analyze the monotonicity, let's find the first derivative: y' = -2cos(2x) On the interval (0,frac{π}{4}), the derivative is negative, meaning that the function is monotonically decreasing. Therefore, the function y=f(x+frac{π}{3}) is an odd function and monotonically decreasing on (0,frac{π}{4}). The correct answer is: boxed{text{D}}.

answer:Solution: 1. Consider a round table with 99 gnomes seated around it. Bilbo the hobbit can ask about the distance between any two gnomes as measured by how many gnomes sit between them along the shortest arc of the circle. Bilbo needs to determine whether there is at least one pair of neighboring gnomes by asking no more than 50 questions. 2. Bilbo starts by focusing on a specific gnome, call him A, and asks about the distance between A and 49 other gnomes. That is, Bilbo will ask 49 questions regarding A: - For example, "How many gnomes sit between A and B?", "How many gnomes sit between A and C?", and so on until he asks about 49 other gnomes. 3. If any answer to these 49 questions is 0, Bilbo has immediately found a pair of neighboring gnomes and his goal is achieved. 4. If Bilbo does not get any answer that is 0, the answers he receives will be numbers between 1 and 48 (the number of gnomes that could potentially sit between any two non-neighboring gnomes). 5. Each response from 1 to 48 can occur at most two times (since there are only two direction arcs in a circle for a given distance). 6. Bilbo can maintain a 2 times 48 array, where each column represents a possible distance (from 1 to 48) and has two rows for recording occurrences: - The top row records the first instance of a distance. - The bottom row records the second instance of the same distance. 7. Given that Bilbo asks 49 questions, there are 98 potential entries (2 rows (times) 48 columns). Since 49 entries will be marked, the Pigeonhole Principle implies that at least one 2 times 2 sub-square will have 3 or 4 of its cells filled: - Each entry in this 2 times 2 sub-square corresponds to a situation where Bilbo knows of three gnomes that potentially form a part of a "trapezoid" (with distances N or N+1 from A). 8. Bilbo then asks the 50th question about the distance between a pair of these gnomes at distances N and N+1 to know if they are neighbors: - Depending on the result of this final question, Bilbo can deduce if they are immediate neighbors, or if their other pair gnome is adjacent to one of them. Conclusively, Bilbo has a strategy that will always allow him to identify at least one pair of neighboring gnomes with no more than 50 questions. Thus, we have: [ boxed{text{Yes}} ]

answer:**Answer** C To solve this problem, we can break it down into steps: 1. Select 1 volunteer for Friday: There are 5 choices. 2. Select 2 volunteers for Saturday from the remaining 4: There are binom{4}{2} = 6 ways. 3. Select 1 volunteer for Sunday from the remaining 2: There are 2 choices. Multiplying these choices together gives us the total number of arrangements: 5 times 6 times 2 = 60. Therefore, the number of different ways to arrange the volunteers is boxed{60} ways, which corresponds to option C.

answer:1. **Determine the total number of slices**: The least common multiple (LCM) of 6, 4, 3, and 7 is 84. Assume the pizza has 84 slices for simplification. 2. **Calculate the slices each sibling ate**: - **Alex**: frac{1}{6} times 84 = 14 slices. - **Beth**: frac{1}{4} times 84 = 21 slices. - **Cyril**: frac{1}{3} times 84 = 28 slices. - **Elsa**: frac{1}{7} times 84 = 12 slices. 3. **Calculate the slices Fred ate**: Total eaten by Alex, Beth, Cyril, and Elsa: [ 14 + 21 + 28 + 12 = 75 text{ slices} ] Therefore, Fred ate: [ 84 - 75 = 9 text{ slices} ] 4. **Order the siblings by the amount of pizza they consumed**: - Cyril ate 28 slices, - Beth ate 21 slices, - Alex ate 14 slices, - Elsa ate 12 slices, - Fred ate 9 slices. The order of consumption from most to least is Cyril, Beth, Alex, Elsa, Fred. **Conclusion**: The correct sequence in descending order is: [ text{Cyril, Beth, Alex, Elsa, Fred} ] The final answer is boxed{textbf{(C)} text{Cyril, Beth, Alex, Elsa, Fred}}