lf `f'(x) > 0,f"(x)>0AA x in (0,1)` and `f(0)=0,f(1)=1`,then prove that `f(x)f^-1(x)

Explanation

Given that `f(x)gt 0,f(x)gt for x in (0,1)`
f(x) is increasing function and concave upward for x in (0,1)
`x in(0,1)`
`f^(-1)(x)` is increasing function and concave downward for `x in (0,1)`
Graphs of y=f(x) and `y=f^(-1)(x)` are as shown in the following figure

From the figure
Slope of OB `gt` Slope of OC
`(x)/sqrt(f(x))gt(f^(-1))(x)/(x)`
`f(x)xxf^(-1)(x)ltx^(2)`