2011/094) Show that the equation 2x+cosx=0 has exactly one real root?

we have
cos x + 2x = 0
so cos x = - 2x

as cos x is between -1 and + 1 so x has to be between -1/2 and + 1/2

now between -1/2 and 1/2

d/dx (cos x + 2x) = 2 - sin x > 0 so increasing

at 1/2 we have cos x + 2x > 0 (both cos x and x > 0)
at -1/2 we have cos x + 2x < 0 cos x < 1 and 2x = - 1

so it is increasing from -1 to 1 and hence exactly one real root between (-1/2 and 1/2)