Let `f:[0,oo)vec[0,oo)a n dg:[0,oo)vec[0,oo)` be non-increasing and non-decreasing functions, respectively, and `h(x)=g(f(x))dot` If `fa n dg` are differentiable functions, `h(x)=g(f(x))dot` If `fa n dg` are differentiable for all points in their respective domains and `h(0)=0,` then show `h(x)` is always, identically zero.

Explanation

Since f(x) `in(0,00),g(f(x))in(0,00)`
Now f(X) is non increasing
`therefore x_(2)gtx_(1)then f(x_(2))lef(x_(1))`
g(x) is non decreasing,
`g(f(x_(2)))leg(f(x_(1)))`
For `xlt0,g(f(x))leg(f(0))`
`therefore g(f(x))le0`
Therefore from, (1) and (2),g(f(x))=0