A polynomial $\varphi \left ( s \right ) = a_{n}s^{n} + a_{n-1}s^{n-1} + \cdots + a_{1}s+a_{0}$ of degree $n>3$ with constant real coefficients $a_{n}, a_{n-1}, \:\dots a_{0}$ has triple roots at $s = -\sigma$. Which one of the following conditions must be satisfied?
- $\varphi \left ( s \right ) = 0$ at all the three values of $s$ satisfying $s^{3}+ \sigma ^{3}=0$
- $\varphi \left ( s \right ) = 0, \frac{d\varphi \left ( s \right )}{ds} = 0$, and $\frac{d^{2}\varphi \left ( s \right )}{ds^{2}} = 0$ at $s = -\sigma$
- $\varphi \left ( s \right ) = 0, \frac{d^{2}\varphi \left ( s \right )}{ds^{2}} = 0$, and $\frac{d^{4}\varphi \left ( s \right )}{ds^{4}} = 0$ at $s = -\sigma$
- $\varphi \left ( s \right ) = 0,$ and $\frac{d^{3}\varphi \left ( s \right )}{ds^{3}} = 0$ at $s = -\sigma$