Question

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f ( x )=\alpha x ^5+\beta x ^3+\gamma x , x \in R$ and $g: R \rightarrow R$ be such that $g(f(x))=x$ for all $x \in R$. If $a_1, a_2, a_3, \ldots, a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n} \displaystyle\sum_{i=1}^n f\left(a_i\right)\right)\right)$ is equal to :

• A. 0
• B. 3
• C. 9
• D. 27

Answer 1

Answer: (A)

Consider a case when $\alpha=\beta=0$ then
$f(x)=y x$
$g(x)=\frac{x}{y}$
$ \frac{1}{ n } \displaystyle\sum_{ i =1}^{ n } f \left( a _{ i }\right) \Rightarrow \frac{ y }{ n }\left( a _1+ a _2+\ldots . .+ a _{ n }\right) $
$ =0 $
$ \Rightarrow f ( g (0)) \Rightarrow f (0) $
$ \Rightarrow 0$