A function `g(x)` is defined as `g(x)=1/4f(2x^2-1)+1/2f(1-x^2) and f(x)` is an increasing function. Then `g(x)` is increasing in the interval. (a) `(-1,1)` (b) `(-sqrt(2/3),0)uu(sqrt(2/3),oo)` (c) `(-sqrt(2/3),sqrt(2/3))` (d) none of these

Options

A. `(-1,1)`
B. `-(sqrt(2)/(3),0) cup (sqrt(2)/(3),oo)`
C. `-sqrt(2)/(3).sqrt(2)/(3)`
D. none of these

Answer

2

Explanation

g(X)=`xf(2x^(2)-1)-xf(1-x)^(2)=xf(2x^(2)-1)-f(1-x^(2))`
`g(x)gt0`
`if xgt0,2x^(21)-1gt1-x^(2)` (has f is na increases function )
`or 3x^(2)gt2 or x in -oo,sqrt(23)/(3)cup sqrt(2)/(3),oo`
`or x in (sqrt(2)/(3),oo)`
if `x lt 0,2 or x in -sqrt(2)/(3),(sqrt(2))/(3)or in -(sqrt(2)/(3),0)`