$If3x^2+4kx+1>0forallrealvaluesofx,thenkliesinthe\mathrm{int}erval$

$\begin{array}{l}Given3x^2+4kx+1>0\\ x^{2}co.eff=3>0\\ x^{2}co.efficentand\exp ressionhavesamesigne\\ \therefore D\le 0\\ \Rightarrow {\left(4k\right)}^2-4\left(3\right)\left(1\right)\le 0\\ \Rightarrow 16k^2-12\le 0\\ \Rightarrow 4k^2-3\le 0\\ \Rightarrow (2k+\sqrt{3})(2k-\sqrt{3})\le 0\\ \Rightarrow k\in [\frac{-\sqrt{3}}{2},\frac{\sqrt{3}}{2}]\end{array}$