Math Problem Solver

Consider function 𝑓(𝑥) = 𝑠𝑖𝑛(2𝑥), for any 𝑥 ∊ 𝑅 (set of all real numbers). In each task (𝑇1),…,(𝑇5) provide a short answer including justification. (𝑇1) Identify all roots of 𝑓(𝑥), i.e., all points 𝑥 ∊ 𝑅 for which 𝑓(𝑥) = 0.

(𝑇2) Identify all intervals (𝑎,𝑏) i.e all points 𝑥 ∊ 𝑅 for which 𝑓(𝑥) is increasing.

(𝑇3) Determine the global maximum value and global minimum value for 𝑑/𝑑𝑥𝑓(𝑥). Also identify all points 𝑥 ∊ 𝑅 for which these extremal values are obtained by 𝑑/𝑑𝑥𝑓(𝑥).

(𝑇4) Identify all intervals (𝑎,𝑏)∊ 𝑅 for which 𝑓(𝑥) is concave.

(𝑇5) What is the nth derivative of 𝑓(𝑥)? Hint: Compute the first few derivatives