Question

Let $f\left(x\right)=a{x}^3+b{x}^2+cx+d\sin x$. If the condition that f(x) is always one-one function is $b^t<ka(c-|d|)$ Find $k+t$

Answer 1

$f\left(x\right)=a{x}^3+b{x}^2+cx+d\sin x$
$f^{\prime }\left(x\right)=3a{x}^2+2bx+c-d\cos x$
For $f\left(x\right)$ to be one-one function,$f^{\prime }\left(x\right)>0\forall x\in R$
$3a{x}^2+2bx+c-d\cos x>0\Rightarrow 3a{x}^2+2bc+c-|d|>0$
$\Rightarrow (2b{)}^2-4.3a(c-|d|)<0\Rightarrow b^2<3a(c-|d|)$