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Find the number of roots of the function...

Find the number of roots of the function `f(x)=1/((x+1)^3)-3x+sinxdot`

Text Solution

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`f(x) =(1)/(x+1)^(3)3x+sin x, x ne-1`
`therefore f(x)=(-3)/(x+1)^(4)-3+cos x=cosx-3(1+(1))/(x+1)^(4)`
Now cos in [-1,1] and `3(1+(1))/(x+1)gt3`
`therefore f(x) lt 0 forall x in R`
So , f(x) is decreasing funtion
Also f(x) is discontious at x=-1
Graph of function is as shown in the following figure.

`therefore` f(x)=0 has roots one in each of the intervals (-00,1) and (-1.00)
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