Question

Let f and g be two differentiable functions on R such that $f^{\prime }\left(x\right)>0$ and $g^{\prime }\left(x\right)<0$, for all $xϵR.$ Then for all $x$ :

• A. $f\left(g\right(x\left)\right)>f\left(g\right(x-1\left)\right)$
• B. $f\left(g\right(x\left)\right)>f\left(g\right(x+1\left)\right)$
• C. $g\left(f\right(x\left)\right)>g\left(f\right(x-1\left)\right)$
• D. $g\left(f\right(x\left)\right)$ $<g\left(f\right(x+1\left)\right)$

Answer 1

Answer: (B)

$f\left(g\right(x\left)\right)>f\left(g\right(x+1\left)\right)$

Let $f\left(x\right)=ax$
$g\left(x\right)=-bx$
where a, b > 0
$f\left(g\right(x\left)\right)\Rightarrow -abx$ ....(1)
$f\left(g\right(x+1\left)\right)\Rightarrow a(-b-bx)$
$=-abx-ab$ .....(2) .
So, from (1) and (2)
$f\left(g\right(x\left)\right)>f\left(g\right(x+1\left)\right)$