Determine the concavity and find the points of inflection.

Question

y=2cosx + sin2x 0≤x≤2pi
y'=-2sinx + 2cos2x
y"=-2cosx-4sinx

How do I find the IP(s)?

Answer 1

Your second derivative's last term should be -4sin2x

set y'' = 0 for points of inflection
-2cosx - 4sin2x = 0
cosx + 2sin2x = 0
cosx + 4sinxcosx = 0
cosx(1 + 4cosx) = 0
cosx = 0 or cosx = -1/4

You seem to know what you are doing, and can probably take it from here.

Concavity in short:
If for some f(x), the second derivative exists,
the curve is concave upwards if f''(x) > 0
the curve is concave downwards if f''(x) < 0
if f''(x) = 0 it is neither and you have a point of inflection as seen above.

Answer 2

from cosx + 4sinxcosx = 0
it should have been
cosx(1 + 4sinx) = 0
cosx = 0 or sinx = -1/4

Answer 3

I got-4sin2x for the second term, I made a typo.

How do you translate sinx=-1/4 to decimal numbers?