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

The Correct Option is A

The correct option is (A) : 0
Consider a case when α = β = 0 then
\(f(x) = yx\)
\(g(x)=\frac{x}{y}\)
\(\frac{1}{n}\sum{^{n}_{i=1}}f(a_i)⇒\frac{y}{n}(a_1+a_2+....+a_n)\)
                          \(=0\)
\(⇒f(g(0))⇒f(0)\)
                     \(⇒0\)