The point of local maximum & minimum of the function f (x) = – x + 2 sin x (where 0 <x<2pi)
The local maximum and minimum points of a function represent the highest and lowest points, respectively, within a particular interval. In the case of the function f(x) = –x + 2sin(x), where 0 < x < 2π, the local maximum and minimum points can be found by identifying the points where the derivative of the function equals zero.
To find the derivative of f(x), we differentiate each term separately:
f'(x) = –1 + 2cos(x)
Setting f'(x) = 0 and solving for x:
–1 + 2cos(x) = 0
2cos(x) = 1
cos(x) = 1/2
From this equation, we can see that x = π/3 and x = 5π/3 (among other solutions), since the cosine of these angles is equal to 1/2.
To determine whether these points are local maximum or minimum, we can analyze the second derivative of f(x). Taking the derivative of f'(x) = –1 + 2cos(x) gives us:
f''(x) = -2sin(x)
Since f''(x) is negative for x = π/3 and positive for x = 5π/3, we can conclude that x = π/3 is a local maximum, and x = 5π/3 is a local minimum, given the constraints of the original function (0 < x < 2π).