Provide an Appropriate Response $f ( x )$ And $g ( x )$

question 193

Essay

Provide an appropriate response.
-Assume that $f ( x )$ and $g ( x )$ are two functions with the following properties: $g ( x )$ and $f ( x )$ are everywhere continuo differentiable, and positive; $\mathrm { f } ( \mathrm { x } )$ is everywhere increasing and $g ( x )$ is everywhere decreasing. Which of the follow functions are everywhere decreasing? Prove your assertions.
i). $h ( x ) = f ( x ) + g ( x )$
ii). $j ( x ) = f ( x ) \cdot g ( x )$
iii). $\mathrm { k } ( \mathrm { x } ) = \frac { \mathrm { g } ( \mathrm { x } ) } { \mathrm { f } ( \mathrm { x } ) }$
iv). $p ( x ) = [ f ( x ) ] g ( x )$
v). $r ( x ) = f ( g ( x ) ) = ( f \circ g ) ( x )$

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