A polynomial ψ(s) = a_nsⁿ + a_n-1s^n-1 + ......+ a₁s + a₀ of degree n > 3 with constant real coefficients a_n, a_n-1, ... a₀ has triple roots at s = -σ. Which one of the following conditions must be satisfied?

The correct option is:
$\psi (s)=0,\frac{d\psi (s)}{ds}=0\text{and}\frac{d^2\psi (s)}{d{s}^2}=0$ at s = -σ

Step-by-step Explanation:

1. Understanding Multiple Roots:
If a polynomial $\psi (s)$ has a root $s=\alpha$ with multiplicity k, it means that $(s-\alpha {)}^k$ is a factor of the polynomial.
Therefore, we can write the polynomial as:
$\psi (s)=(s-\alpha {)}^k\cdot Q(s)$
where $Q(s)$ is a polynomial such that $Q(\alpha )\ne 0$.

For a triple root at $s=-\sigma$, the multiplicity is k = 3. Substituting this into our expression yields:
$\psi (s)=(s+\sigma {)}^3\cdot Q(s)$
Evaluating this at $s=-\sigma$:
$\psi (-\sigma )=(-\sigma +\sigma {)}^3\cdot Q(-\sigma )=0$

2. Finding the First Derivative:
Let us find the derivative of $\psi (s)$ with respect to s using the product rule:
$\frac{d\psi (s)}{ds}=3(s+\sigma {)}^2\cdot Q(s)+(s+\sigma {)}^3\cdot Q^{\prime }(s)$
Factoring out $(s+\sigma {)}^2$:
$\frac{d\psi (s)}{ds}=(s+\sigma {)}^2[3Q(s)+(s+\sigma )Q^{\prime }(s)]$
Evaluating this at $s=-\sigma$:
${\left[\frac{d\psi (s)}{ds}\right]}_{s=-\sigma }=(-\sigma +\sigma {)}^2[3Q(-\sigma )+0]=0$

3. Finding the Second Derivative:
Now let us find the second derivative of $\psi (s)$ with respect to s by differentiating the first derivative expression again:
$\frac{d^2\psi (s)}{d{s}^2}=2(s+\sigma )[3Q(s)+(s+\sigma )Q^{\prime }(s)]+(s+\sigma {)}^2\cdot \frac{d}{ds}[3Q(s)+(s+\sigma )Q^{\prime }(s)]$
Notice that every term contains at least one factor of $(s+\sigma )$.
Evaluating this at $s=-\sigma$:
${\left[\frac{d^2\psi (s)}{d{s}^2}\right]}_{s=-\sigma }=0+0=0$

4. Analyzing the Third Derivative:
If we were to take the third derivative, it would result in:
${\left[\frac{d^3\psi (s)}{d{s}^3}\right]}_{s=-\sigma }=3!\cdot Q(-\sigma )=6Q(-\sigma )\ne 0$
Thus, the condition for a triple root at $s=-\sigma$ is precisely that the function value, its first derivative, and its second derivative must all vanish at that point, while the third derivative remains non-zero.