The number of solution x of the equation `sin (x+x^(2))- sin(x^(2))= sin x` in the interval [2,3] is

Options

A. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

`2 cos(x+(2x^(2))/(2)).sin ""(x)/(2)= 2 sin ""(x)/(2). cos ""(x)/(2)`
`implies sin ""(x)/(2)[cos((x+2x^(2))/(2))= cos "" (x)/(2)]=0`
`sin ""(x)/(2)=0 " " or " "2 sin((2x+2x^(2))/(4)).sin((2x^(2))/(4))=0`
`(x)/(2)=0,pi,2pi " "or " "sin((x+x^(2))/(2))=0 " "or" " sin ""(x^(2))/(2)=0`
`X=0,2pi,4pi " " or " " (x+x^(2))/(2)=0,pi,2pi`
`(x+x^(2))/(2)=pi " " x^(2)=2pi`
`implies x^(2)+x-2pi=0" " impliesx=sqrt(2pi)`
`x=(-1pmsqrt(1+8pi))/(2)" "2 gtsqrt(2pi) lt3`
`sqrt(1+8pi)~~sqrt(25.14)`
`" "~~5.2`
`:. x=(5.2-1)/(2)=(4.2)/(2)=2.1`
`:.` total number of solution lies between (2,3)=2