If Dx x 2cosx represents the distance traveled by a partic

If D(x) = x + 2cosx represents the distance traveled by a particle from x = 0 to x = 2pi . Find an x value in (0, 2pi ) for which the instantaneous velocity equals the average velocity.

v=d(D(x)/dx=1-2sinx a=d(d(D(x)/dx)/dx=-2cosx 1-2sinx=-2cosx 2cosx-2sinx=1 cosx-sinx=1/2 multiply by 1/sqrt2 both sides cos(pi/4)cosx-sin(pi/4)sinx=1/2sqrt2 cos(pi/4+x)=1/2sqrt2 x=cos^-1(1/2sqrt2)-pi/4=0.4238