Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)

Question

Answer 1

f(x) = x + 2sinx

what are decreasing & increasing intervals for 0 <x< 2pi??

take the derivative, set = 0 to get end points of the intervals

f'(x) = 1 +2cosx = 0

cosx = -1/2

x = arccos-.5 = 120 degrees and 240 degrees

= 4pi/3 and 2pi/3

that gives 4 intervals

(0, 2pi/3), (2pi/3, 2pi), (2pi, 4pi/3) and (4pi/3, 2pi)

1st & last are increasing

middle 2 are decreasing

f'(0) = 1 >0

f'(2pi) = 1 >0

f'(pi) = 0 =0

Answer 2

f(x) = x + 2sinx

f'(x) = 1 + 2cosx f'(x) = 0 when cosx = -1/2. x = 2π/3, 4π/3

When 0 < x < 2π/3, f'(x) > 0. So, f is increasing

When 2π/3 < x < 4π/3, f'(x) < 0. So, f is decreasing.

When 4π/3 < x < 2π, f'(x) > 0. So, f is increasing.