Question

Prove that the equation `2 sin x=|x|+a` has no solution for `a in ((3sqrt(3)-pi)/3, oo)`.

Answer 1

`sinx=(1)/(2)|x|+(a)/(2)`
Draw the graphs of `y=2sinxandy=|x|`.

The equation `2sinx=|x|+a` will have a solution so long as the line `y=|x|+a` intersects or at least touches the curve `y=2sinx`. When the graphs touch, we have
`(dy)/(dx)=2cosx=1,impliesx=(pi)/(3)`
Hence for no solution,
`(pi)/(3)+agt2sin""(pi)/(3)`
`impliesagt(3sqrt3-pi)/(3)`