Question

Sketch
f (x) = x + cos x on
[-2pie,2pie ]. Find any local extrema, inflection points, or asymptotes. And find the absolute maximum and absolute minimum values of f on the given interval.

i cant seem to figure out how to solve this

Answer 1

for local extrema f'(x) = 0
1 - sinx = 0
sinx=1
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
-cosx=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.