f(x) = cos^2x - 2sinx

f'(x) = -2cosx*sinx - 2cosx

f''(x) = -2cos^2x + 2sin^2x + 2sinx

Set f'(x) = 0

0 = -2cosx*sinx - 2cosx

0 = -2cosx(sinx + 1)

cosx = 0 --> x = pi/2, 3pi/2

sinx + 1 = 0 --> sinx = -1 --> x = 3pi/2

The roots of f'(x) are: x = pi/2 and x = 3pi/2

Choose values between the following intervals to find where the graph is increasing or decreasing:

For (0,pi/2), choose x = pi/4 --> f'(pi/4) = (negative)

For (pi/2,3pi/2), choose x = pi --> f'(pi) = (positive)

For (3pi/2,2pi), choose x = 7pi/4 --> f'(7pi/4) = (negative)

Decreasing: (0,pi/2) U (3pi/2,2pi)

Increasing: (pi/2,3pi/2)

Minimum: x = pi/2, x = 2pi

Maximum: x = 3pi/2, x = 0

Set f''(x) = 0

0 = -2cos^2x + 2sin^2x + 2sinx

0 = -2(1 - sin^2x) + 2sin^2x + 2sinx = -2 + 4sin^2x + 2sinx

0 = (sinx + 1)(4sinx - 2)

sinx + 1 = 0 --> sinx = -1 --> x = 3pi/2

4sinx - 2 = 0 --> sinx = 1/2 --> x = pi/6, 5pi/6

The roots of f''(x) are: x = 3pi/2, x = pi/6, x = 5pi/6

Choose values to find concavity:

For (0,pi/6), choose x = pi/12 --> f''(pi/12) = (negative)

For (pi/6,5pi/6), choose x = pi/2 --> f''(pi/2) = (positive)

For (5pi/6,3pi/2), choose x = pi --> f''(pi) = (negative)

For (3pi/2,2pi), choose x = 7pi/4 --> f''(7pi/4) = (negative)

Concave Up: (pi/6,5pi/6)

Concave Down: (0,pi/6) U (5pi/6,3pi/2) U (3pi/2,2pi)

Inflextion: x = pi/6, x = 5pi/6

Hopefully I made no mistakes in all of that. I suggest you check my work.