Question

Let $f$ and $g$ be two differentiable functions on $\mathbb{R}$ such that $f'(x) > 0$ and $g'(x) < 0$, for all $x \in \mathbb{R}$. Then for all $x$:
A. $f(g(x)) > f(g(x - 1))$
B. $f(g(x)) > f(g(x + 1))$
C. $g(f(x)) > g(f(x - 1))$
D. $g(f(x)) < g(f(x + 1))$

Answer 1

Here two functions $f$ and $g$ are given. So, from the given information, we will first determine which function is increasing and which function is decreasing. Then, we are to analyse the options to find which one is the correct option. To analyse the options, the concept of composition of two functions is to be used. Following the above steps we can answer the given problem.

Complete step by step answer:
It is given that, $f'(x) > 0$ and $g'(x) < 0$. We know, if the first derivative of a function is greater than$0$, then it is an increasing function. Also, if the first derivative of a function is smaller than$0$, then it is a decreasing function. So, from the given conditions we can say that, $f$ is an increasing function, i.e. $f(x - 1) < f(x) < f(x + 1),\forall x \in \mathbb{R}$ since we are given that $f'(x) > 0$. Also, g is a decreasing function, i.e. $g(x - 1) > g(x) > g(x + 1),\forall x \in \mathbb{R}$ since we are given that $g'(x) < 0$.

Now, we analyse options one by one.
Analysing option A:
We are given that: $f(g(x)) > f(g(x - 1))$
We know that, $g(x) < g(x - 1)$
Now, by composition of two functions,
$f(g(x)) < f(g(x - 1))$
$\left[ {\because f(x) > f(x - 1){\text{ and }}g(x) < g(x - 1)} \right]$
So, option A is not correct.

Analysing option B:
We are given that: $f(g(x)) > f(g(x + 1))$
We know that, $g(x) > g(x + 1)$
Now, by composition of two functions,
$f(g(x)) > f(g(x + 1))$
$\left[ {\because f(x) < f(x + 1){\text{ and }}g(x + 1) < g(x)} \right]$
So, option B is correct.

Analysing option C:
We are given that: $g(f(x)) > g(f(x - 1))$
We know that, $f(x) > f(x - 1)$
Now, by composition of two functions,
$g(f(x)) < g(f(x - 1))$
$\left[ {\because g(x) < g(x - 1){\text{ and }}f(x) > f(x - 1){\text{ }}} \right]$
So, option C is not correct.

Analysing option D:
We are given that: $g(f(x)) < g(f(x + 1))$
We know that, $f(x + 1) > f(x)$
Now, by composition of two functions,
$g(f(x)) > g(f(x + 1))$
$\left[ {\because g(x) > g(x + 1){\text{ and }}f(x + 1) > f(x){\text{ }}} \right]$
So, option D is not correct.

Therefore, the only correct option is option B.

Note: In this question, the first derivative of functions was given, so it was easy to determine whether the function is increasing or decreasing. But, in some questions, sometimes, the function itself is given. In such problems, either the function can be differentiated and whether the function is increasing or decreasing can be determined, or the function can be analysed and it’s trend can also be analysed, which is sometimes done by finding the values of the functions for simple terms like $0$ and $1$.