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Show that tan^(-1)x > x/(1+(x^2)/3)ifx i...

Show that `tan^(-1)x > x/(1+(x^2)/3)ifx in (0,oo)dot`

Text Solution

Verified by Experts

Let us assume f(x) =`tan^(-1)x-(3x)/(x^(2)+3)`
`therefore f(x)=(1)/(1+x^(32))-(3(x^(2)+3)-3x(2x))/(X^(2)+3)^(2)`
`=(4x^(4))/((1+x^(2))(x^(2)+33)^(2))gt forall x in (0,oo)`
Hence f(x) is increasing throughout
Also f(x)=0
Hence `f(X) gt0 forall x gt0`
or `tan^(-1)xgt(3x)/(3+x^(2))`
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