Let `g(x)=f(x)+f(1-x)` and `f''(x)>0AAx in (0,1)dot` Find the intervals of increase and decrease of `g(x)dot`

Explanation

We have g(x) =f(x) + f(1-x)
g(x) =f(x)-f(1-x)
Given that `f(x)gt0,forall` x in(0,1)
It means that f(x) is increasing on (0,1)
Now g(x) is increasing.
`therefore f(x)-f(1-x)gt0`
`rArr f(x)gtf(1-x)`
`rArr xgt1`-x As f(x) is increasing
`rArr (1)/(2)ltxlt1`
So g(x) is increasing in`(1)/(2), 1`
If g(x) is decreasing
`therefore f(x)ltf(1-x)`
`rArr xlt1`-x As f(x) is increasing
`rArr 0ltxlt(1)/(2)`
So g(x) is decreasing in `0,(1)/(2)`