SOLUTION: Find the stationary point on the curve y = sinx-0.5x for o < x < π and justify that it is a local maximum.

You are given a function  y = sin(x) - 0.5x.


Its stationary point is the point x, where y'(x) = 0,  or

    y'(x) = cos(x) - 0.5 = 0,

            cos(x) = 0.5.


In the given interval  0 < x < π,  there is only one such point: it is the point  x = 60 degrees,  or  x =  radians
(the value about   = 1.046...).


See the plot of the function in the Figure below


    


                 Plot  y = sin(x) - 0.5x



From the plot, you see that the curve in vicinity of  x =  = 1.046...  is  like a parabola opened down.


The second derivative  y''(x) = -sin(x)  has negative value  y''(pi/3) =  =   at this point.


It justifies that the stationary point  x=   is the local maximum of the function y = sin(x) - 0.5x.