Let `g(x)=2f(x/2)+f(2-x)a n df^(x)<0AAx in (0,2)dot` Then `g(x)` increases in `(1/2,2)` (b) `(4/3,2)` `(0,2)` (d) `(0,4/3)`

Options

A. increasing on (4/3, 2) and increasing on (0,4/3)
B. decreasing on (0,4/3) and decreasing on (4,3,2)
C. increasing (0,4/3) and decreasing on (4,3,2)
D. non of these

Answer

C

Explanation

we have `g(x)=2f(x/2)+f(2-x)for all x in (0,2)`
`rArr g(x)=f'(x) -f'(2-x) for all x in (0,2)`
Now lt brgt `f'(x/2) gt f'(2-x),if x/2 lt 2-x`
`rArr f(x/2)lt (2-x)if x lt4/3 and f'(x/2) gt f'(2-x),ifx gt 4/3`
`rArr g'(x) gt 0, if x in (0,4//3)` and decreasing on (4/3,2)