Problem- Brutus is working on a new space telescope-called Scarlet Eye-to help detect exoplanets. The telescope works by focusing on one star and recording the star's brightness. (Brutus hopes to locate exoplanets by measuring the differences in a star's brightness. If the brightness significantly decreases at regular intervals, then its change in brightness is probably caused by a planet passing in front of the star.)

Scarlet Eye is launched into space and starts returning data on its first star. Unfortunately, Brutus discovers that before the launch Sparty has hacked the telescope! Sparty's hack forces the telescope to only return functions and not their graphs.

The telescope returns

f (x) = 3sin (x/2)

for its first star. Undeterred Brutus's knows calculus and so knows how to convert this function into its graph.

The Scarlet Eye tells Brutus that the domain of this function is [-2π, 2π]. (Brutus knows that usually Brutus should always figure out the domain of the function first.)

(A)

(i) True or False: this function is an even function. (ii) True or False: this function is an odd function.

(B) Find the first derivative of f.

(C) Find all critical points of f.

(D) Identify those intervals where f is increasing and those intervals where f is decreasing.

(E) Use the first derivative test to identify local extrema.

(F) Find the second derivative of f.

(G)

(i) Identify those intervals where f is concave up and those intervals where f is concave down. (ii) Find all the inflection points of f

(H) Sketch the graph of f over the interval [-2π, 2π]. In your graph be sure to mark critical points, inflection points, local extrema, global extrema, and x-intercepts and y-intercepts.