Question

If $f\text{'}\left(x\right)>0$ and $g\text{'}\left(x\right)<0,x\in R$ then

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

Answer 1

The correct option is A

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

Explanation for the correct option:

Finding the value.

Let $f\left(x\right)=ax$

$g\left(x\right)=-bx$, where $a,b>0$

Now,

$f(g(x\left)\right)=-abx$ ….(1)

$f(g(x+1\left)\right)=a(-b-bx)$

$=-abx-ab$ .... (2)

compare equation (1) and (2) , we get

$\mathit{f}\mathbf{}\mathbf{\left(}\mathit{g}\mathbf{\right(}\mathit{x}\mathbf{\left)}\mathbf{\right)}\mathbf{}\mathbf{>}\mathbf{}\mathit{f}\mathbf{}\mathbf{(}\mathit{g}\mathbf{}\mathbf{(}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{\left)}\mathbf{\right)}$

Hence, Option ‘A’ is Correct.