Question

If the real-valued function $f\left(x\right)=x^3+3(a^2-1)x+1$ be invertible then the set of possible real values of a is

• A. $(-\infty ,-1)\cup (1+\infty )$
• B. $(-1,1)$
• C. $[-1,1]$
• D. $(-\infty ,-1]\cup [1,+\infty )$

Answer 1

The correct option is D $(-\infty ,-1]\cup [1,+\infty )$

Given,

$f\left(x\right)=x^3+3(a^2-1)x+1$
$f^{\prime }\left(x\right)=3x^2+3(a^2-1)$

Now for $f\left(x\right)$ to be invertible it has to be bijective and for that $f\left(x\right)$ should be increasing in this case,
i.e $f^{\prime }\left(x\right)>0\forall x\in R\Rightarrow a^2-1>0$

$\Rightarrow a\in (-\infty ,0)\cup (1,\infty )$