If `f(x)=x/(sinx) \ a n d \ g(x)=x/(tanx),w h e r e \ 0ltxlt=1,` then in this interval

Options

A. both f(x) and g(x) are increasing functions
B. both f(x) and g(x) decreasing functions
C. f(x) is an increasing function
D. g(x) is an increasing function

Answer

C

Explanation

We have
`f(x) =x/(sinx) "and" g(x) = x/(tanx)`
`therefore f'(x) = (sinx-xcosx)/(sin^2x) " and " g'(x) = (tanx-xsec^2x)/(tan^2x)`
Let `phi(x) = sinx=-xcosx " and " psi(x) = tanx-xsec^2x`
Then, `f(x) = (phi(X))/(sin^2x) " and " g'(x) = (psi(x))/(tan^2x)`
Now,
`phi(x)=cosx-cosx+xsinx=xsinx`
and,
`phi(x)=sec^2x-sec^2-2xsec^2tanx=-2xsec^2xtanx`
For` 0 lt xx le 1` we have
`xgt0, sinxgt0, tanxgt0,secxgt0. `
`rArr phi(x)=xsinxgt0and psi (x) lt0 " for " 0 ltx le 1`
`rArr phi'(x) " is increasing on (0,1]and " psi(x) ` is discreasing on (0,1]
`phi(x)gt" 0 " phi(0) " and " psi(x) ltpsi " 0 " `
`rArr phi(x)gt0 " and " psi (x) lt0`
`rArr f'(x)= (phi(x))/(sin^2x)gt" 0 and " g'(x) = (phi(x))/(tan^2x)lt 0`
`rArr` is increasing on(0,1] and g(x) is decreasing on (0,1]