Let $$f$$ and $$g$$ be functions from the interval $$\left[ {0,\,\infty } \right)$$  to the interval $$\left[ {0,\,\infty } \right),\,f$$   being an increasing and $$g$$ being a decreasing function. If $$f\left\{ {g\left( 0 \right)} \right\} = 0$$   then :

Solution :
$$\eqalign{ & f'\left( x \right) > 0{\text{ if }}x \geqslant 0{\text{ and }}g'\left( x \right) < 0{\text{ if }}x \geqslant 0 \cr & {\text{Let }}h\left( x \right) = f\left( {g\left( x \right)} \right){\text{ then}} \cr & h'\left( x \right) = f'\left( {g\left( x \right)} \right).g'\left( x \right) < 0{\text{ if }}x \geqslant 0 \cr & \therefore \,h\left( x \right){\text{is decreasing function}} \cr & \therefore \,h\left( x \right) \leqslant h\left( 0 \right){\text{ if }}x \geqslant 0 \cr & \therefore \,f\left( {g\left( x \right)} \right) \leqslant f\left( {g\left( 0 \right)} \right) = 0 \cr & {\text{But codomain of each function is }}\left[ {0,\,\infty } \right) \cr & \therefore \,f\left( {g\left( x \right)} \right) = 0{\text{ for all }}x \geqslant 0 \cr & \therefore \,f\left( {g\left( x \right)} \right) = 0 \cr & {\text{Also }}g\left( {f\left( x \right)} \right) \leqslant g\left( {f\left( 0 \right)} \right){\text{ }}\left[ {{\text{as above}}} \right] \cr} $$