Prove that
`3 tan^(-1) x= {(tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " -(1)/(sqrt3) lt x lt (1)/(sqrt3)),(pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x gt (1)/(sqrt3)),(-pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x lt - (1)/(sqrt3)):}`

Options

A. `f'(3)=(3)/(1+x^(2))"for all x"in R-{(-1)/(sqrt3),(1)/(sqrt3)}`
B. `f'(x)=(3)/(1+x^(2))"for all "x in R`
C. f(x) is not differentiable at infinitely many points.
D. none of these

Answer

A

Explanation

We have
`f(x)=tan^(-1) ((3x-x^(3))/(1-3x^(2)))`
`f(x)={{:(,3tan^(-1)x,"if "-(1)/(sqrt3)lt x lt (1)/(sqrt3)),(,3tan^(-1)x-pi,"if "gt (1)/(sqrt3)),(,3tan^(-1)x+pi,"if "x lt (-1)/(sqrt3)):}`
`f'(x)={{:(,(3)/(1+x^(2)),"if "-(1)/(sqrt3)lt x lt (1)/(sqrt3)),(,(3)/(1+x^(2)),"if "x lt (1)/(sqrt3)),(,(3)/(1+x^(2)),"if "x lt -(1)/(sqrt3)):}`
`Rightarrow f'(x)=(3)/(1+x^(2))"for all "x in R-{(-1)/(sqrt3),(1)/(sqrt3)}`