Test Index

Functions Part 1

Section: Mathematics

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Question : 11 of 100

Marks: +1, -0

Let

\alpha, \beta

\gamma

be three positive real numbers. Let

f(x)=\alpha x^5+\beta x^3+\gamma x, x\in\mathbb{R}

g:\mathbb{R}\rightarrow\mathbb{R}

g(f(x))=x

x\in\mathbb{R}

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}\sum\limits_{i=1}^{n} f(a_i)\right)\right)

[28-Jul-2022-Shift-1]

0
3
9
27

Solution:

f\left(g\left(\;\frac{1}{n}\sum\limits_{i=1}^{n} f(a_i)\right)\right)

\;\frac{a_1+a_2+a_3+\cdots+a_n}{n}=0

\therefore

First and last term, second and second last and so on are equal in magnitude but opposite in sign.

f(x)=\alpha x^5+\beta x^3+\gamma x

\sum\limits_{i=1}^{n} f(a_i)=\alpha(a_1^5+a_2^5+a_3^5+\cdots+a_n^5)+\beta(a_1^3+a_2^3+\cdots+a_n^3)+\gamma(a_1+a_2+\cdots+a_n)

=0\alpha+0\beta+0\gamma

=0

\therefore f\left(g\left(\;\frac{1}{n}\sum\limits_{i=1}^{n} f(a_i)\right)\right)=\;\frac{1}{n}\sum\limits_{i=1}^{n} f(a_i)=0

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