Question

Find all relative extrema and points of inflection of the function:

f(x) = sin (x/2)

0 =< x =< 4pi

=< is supposed to be less than or equal to.

I can find the extrema, but the points of inflection has me stumped. The inflection point is (2pi,0) but shouldn't there be inflection points on x=0 and x=4pi as well?

Answer 1

The basic sine curve has points of inflection at every x-intercept of the curve,

y' = (1/2) cos x/2
y' = -(1/4) sin x/2

-(1/4)sin x/2 = 0
gives me x/2 = 0 or x/2 = π or x/2 = 2π etc
or
x = 0 or x = 2π or x = 4π etc

then if you domain is 0 ≤ x ≤ 4π
you are correct, there should have been points of inflection at
(0,0), (2π,0) and (4π,0)