Question:
If $f\left(x\right) = \frac{a^{2} -1}{a^{2} +1} x^{3} -3x +5 $ is a decreasing
function of $x$ in $R$, then the set of possible values of a (independent of $x$) is
If $f\left(x\right) = \frac{a^{2} -1}{a^{2} +1} x^{3} -3x +5 $ is a decreasing
function of $x$ in $R$, then the set of possible values of a (independent of $x$) is
Updated On: Jul 28, 2022
- $(1, \infty)$
- $(- \infty , - 1 )$
- $[-1,1]$
- None of these
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The Correct Option is C
Solution and Explanation
$f'(x)=3\left(\frac{a^{2}-1}{a^{2}+1}\right) x^{2}-3$
$f'(x) < 0$ for all $x $ if $a^{2}-1 \leq 0$
$ \Rightarrow-1 \leq a \leq 1$
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Concepts Used:
Types of Functions
Types of Functions
One to One Function
A function is said to be one to one function when f: A → B is One to One if for each element of A there is a distinct element of B.
Many to One Function
A function which maps two or more elements of A to the same element of set B is said to be many to one function. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function.
Read More: Types of Functions