Question

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

Answer 1

Given that $f\left(x\right)=x^{3}+ax^{2}+5x+sin2x$
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.
$\frac{d y}{d x}=3x^{2}+2ax+5+2cos2x>0$
$\Rightarrow 3x^{2}+2ax+5>-2cos2x$
$\Rightarrow 3x^{2}+2ax+5>2$ $\left(\because cos 2 x \in \left[- 1 , 1\right]\right)$
$3x^{2}+2ax+3>0$
If $ax^{2}+bx+c>0\forall x\in R\Rightarrow a>0,b^{2}-4ac < 0.$
$D < 0$
$\Rightarrow 4a^{2}-4\times 3\times 3 < 0$
$\Rightarrow \left(a - 3\right)\left(a + 3\right) < 0$
$\Rightarrow -3 < a < 3$