Question

find the points of inflection and discuss the concavity of the graph of this function. 2csc3x/2, with the points of (0,2pie)

Answer 1

ok to evaluate concavity we need the 2nd derivative
what is derivative of csc ?

Answer 2

here is graph
http://www.wolframalpha.com/input/?i=f%28x%29+%3D+2csc%283x%2F2%29+for+x+%3D0+to+2pi

notice csc is undefined when sine is 0
sin is 0 when angle is multiple of pi

3x/2 = npi

x = n*(2pi/3)

so from (0,2pi) the vertical asymptotes occur when
x = 0
x = 2pi/3
x = 4pi/3
x = 2pi

from graph you can see that its concave up when
0<x< 2pi/3 and 4pi/3 < x < 2pi

concave down when
2pi/3 < x < 4pi/3

hope that helps