Consider the function given by f(x) = e/2-earctan (1+x²). (Note: arctan (1+x²) is the power to which e is being raised.)

[answerhappygod]

Consider The Function Given By F X E 2 Earctan 1 X Note Arctan 1 X Is The Power To Which E Is Being Raised 1 (141.81 KiB) Viewed 9 times

Consider the function given by f(x) = e/2-earctan (1+x²). (Note: arctan (1+x²) is the power to which e is being raised.) This function has derivatives: f'(x) 2xearctan(1+x²) (x² + 1)² +1' 2(3x4-2)earctan (1+x²) ((x² + 1)² + 1)² 1. (4 points) What is the domain of f? (Give your answer in simplified interval notation). 2. (4 points) Build a sign chart for f'(x). 3. (6 points) Build a sign chart for f"(x). 4. (6 points) On which interval(s) is f BOTH increasing AND concave up? Justify your answer by citing relevant informa- tion from questions 2) and 3). 5. (6 points) On which interval(s) is f BOTH decreasing AND concave up? Justify your answer by citing relevant informa- tion from questions 2) and 3). 6. (6 points) On which interval(s) is f BOTH increasing AND concave down? Justify your answer by citing relevant infor- mation from questions 2) and 3). f"(x) 7. (6 points) On which interval(s) is f BOTH decreasing AND concave down? Justify your answer by citing relevant infor- mation from questions 2) and 3). 8. (6 points) Find the x-values of any local extrema. State whether it is a local maximum or minimum, and explain how to determine this based on the sign charts above. Be sure to find the output value of the function at those points. = 9. (6 points) Find the x-values of any inflection points. Explain what conditions need to be checked to verify that these are inflection points. 10. (10 points) Assume that lim f(x) = 0 and lim_ f(x) = 0. x-x -5 Sketch a graph of f based on your work above. Be sure to identify and label any local extrema and inflection points on the graph. <-4 -3 -2 -1 1 2 3 4 5