Question

The function $f\left(x\right)=x^3+\lambda x^2+5x+\sin 2x$ will be an invertible function if $\lambda$ belongs to

• A. $(-\infty ,-3)$
• B. $(-3,3)$
• C. $(3,+\infty )$
• D. none of these

Answer 1

The correct option is B $(-3,3)$

$f\left(x\right)=x^3+\lambda x^2+5x+\sin 2x$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+2\lambda x+5+2\cos 2x$
For $f\left(x\right)$ to be invertible $f^{\prime }\left(x\right)>0\forall x\in R$
$3x^2+2\lambda x+5+2\cos 2x>0\Rightarrow 3x^2+2\lambda x+3>0$
For this discriminant of above quadratic should be negative,
$\Rightarrow {\lambda }^2-9<0$

$\Rightarrow \lambda \in (-3,3)$