Discuss the monotonocity of `Q(x),` where `Q(x)=2f((x^2)/2)+f(6-x^2)AAx in R` It is given that `f^(x)>0AAx in Rdot` Find also the point of maxima and minima of `Q(x)dot`

Explanation

Give g(x) =`2f(x^(2))/(2)+f(6-x)`
`g(x)=2xy(x^(2))/(2)-2x(f(6-x^(2)))`
`=2x(f(x^(2)))/(2)-f(6-x^(2))`
But given that `f(x)gt0`
So f(x) is increasing for all x `in` R
Let g(x)=0
`therefore 2x(f(x^(2)))/(2)-f'(6-x^(2))=0`
x=0 or `f(x^(2))/(2)=f(6-x^(2))`
`x=0 "or" (x^(2))/(2)=6-x^(2)`
`x=0 "or" x^(2)=4`
`x= "or" x^(2)= pm 2`
sign scheme of g(x) is as follows

From the sign schem of g(x) ,g(x) increases in `(-2,0)cup(2,oo)` and decreases in `(-oo,-2)cup(0,2)`
Also x =-2 and x =2 are points of minima and x=0 is point of maxima.