Answer:
2.598 and -2.598.
Explanation:
f(x) = 2 cos x + sin 2x
f'(x) = -2 sin x + 2 cos 2x = 0 for turning points.
cos 2x = 1 - 2 sin^2 x so we have
-2 sin x + 2 - 4 sin^2 x = 0
4sin^2 x + 2 sin x - 2 = 0
2(2 sin^2 x + sin x - 1) = 0
2(2sinx - 1)(sinx + 1) = 0
sin x = 0.5, -1 when f(x) is at a turning point.
x = π/6, -π/2, 5pi/6
The second derivative is 2 cos x + 2 * -2 sin 2x
= 2 cos x - 4 sin 2x
When x = π/6, this is negative , when x = -π/2 it is positive
so x = π/6 gives a maximum f(x) and x = -π/2 gives 0 so this is a point of inflection
When x = π/6 , f(x) = 2.598
When x = 5pi/6, f(x) = -2.598.