Condition for 2 real and distinct roots of a cubic equation

Question

Determine the exact value of $k$ if the cubic equation $x^3 + kx +4 = 0$ has 2 distinct real roots

Answer 1

Let $f(x)=x^3+kx+4$. Being a real cubic equation, $f(x)=0$ will have at least one real root. It will have other roots real if $f_{max}>0$ and $f_{min}<0$. It will have one root as double root if $f_{max}f_{min}=0.$ Here $f'(x)=0 \implies x=\pm \sqrt{-k/3}$. Then $f_{max}=f(-\sqrt{-k/3})=4-\frac{2k}{3}\sqrt{-k/3}, f_{min}=f(-\sqrt{-k/3})=4+\frac{2k}{3}\sqrt{-k/3}.$ Next, $f_{min}f_{max}=0 \implies k=-3(4)^{1/3}.$

So when $k=-3(4)^{1/3}=-4.7622$, apart from one essential real root there will be one double root. Consequently for this special value of $k$, $f(x)=0$ will have two distinct real roots. S below see the plot of $f(x)$