Question

If the equation $4x^3+5x+k=0$(k $\in$R) has a negative real root then k will be

Answer 1

Let $f\left(x\right)=4x^3+5x+k$
$f^{\prime }\left(x\right)=12x^2+5>0\forall x\in R$
$\therefore$ f(x) is strictly increasing on R.
So, for $f\left(x\right)=0$ to have a negative real root, $f\left(0\right)>0\Rightarrow k>0$.