for local extrema f'(x) = 0
1 - sinx = 0
x = pi/2 or -3pi/2 for the given domain
f(pi/2) = pi/2 + cos(pi/2) = pi/2 + 0 = pi/2
for x=-3pi/2 f(-3pi/2) = -3pi/2
so the local extrema are pi/2 and -3pi/2
inflection points : f''(x) = 0
x = ±pi/2 or ±3pi/2
sub those x values back into original to get the y of the points of inflection
consider the end points of the domain for absolute max/mins
f(2pi) = 2pi + cos(2pi) = 2pi + 1
f(-2pi) = -2pi + 1
compare these with the local max/mins and use your calculator to determine the largest and smallest values
BTW, no asymptotes, the graph will be a cosine curve rising as x gets larger.
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