Question

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

• A. $g\left(x\right)\ge g\left(0\right)$
• B. $g\left(x\right)\le g\left(0\right)$
• C. $g\left(2\right)=0$
• D. none of these

Answer 1

Answer: (B), (C)

$g\left(x\right)\le g\left(0\right)$
$g\left(2\right)=0$

For $x\ge 0$ we have $g\left(x\right)\le g\left(0\right)$ as the function $g$ is a decreasing function.
but then $f\left(g\left(x\right)\right)\le f\left(g\left(0\right)\right)=0$ as the function $f$ is an increasing function.
Hence we have for $x\ge 0$ and $f\left(g\left(x\right)\right)\le 0$
i.e., $f\left(g\left(x\right)\right)=0$ for all $x\ge 0$ as $0$ is smallest value $f$ can attend. and hence $g\left(x\right)=0$ for all $x$. In particular $g\left(2\right)=0$