Question

If the function $f:R\to R$ given by $f\left(x\right)=x^3+a{x}^2+5x+sin2x$ is invertible and $a\in \left(-3,1\right)$ then find $\lambda$ .

Answer 1

Answer: 3

Given that $f\left(x\right)=x^3+a{x}^2+5x+sin2x$
Function inverse exists means it should be bijective i.e one-one and onto.
We knows that if a function is one-one is also strictly increasing function.
$\frac{dy}{dx}=3x^2+2ax+5+2cos2x>0$
$\Rightarrow 3x^2+2ax+5>-2cos2x$
$\Rightarrow 3x^2+2ax+5>2$ $\left(\because cos2x\in \left[-1,1\right]\right)$
$3x^2+2ax+3>0$
If $a{x}^2+bx+c>0\forall x\in R\Rightarrow a>0,b^2-4ac<0.$
$D<0$
$\Rightarrow 4a^2-4\times 3\times 3<0$
$\Rightarrow \left(a-3\right)\left(a+3\right)<0$
$\Rightarrow -3<a<3$