What is the main relationship between the inflection points and the roots?

Question

How does inflection point equal the average of real parts of three roots in a cubic equation and what does it refer to in nth-degree polynomial equations?

Answer 1

We will show algebraically that the inflection point, $a$, of a cubic equation with three real roots is the average of the roots.

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Assume a cubic has three real roots, $r_1, r_2$, and $r_3$. Then it is expressible in the following form:

$$c\cdot(x-r_1)(x-r_2)(x-r_3)$$ $$= c\cdot\left(x^3 - (r_1+r_2+r_3)x^2 + (r_1 r_2+r_1 r_3+r_2 r_3)x - r_1 r_2 r_3\right)$$

The inflection point, $a$, is the point at which the second derivative equals zero.

$$f''(a) = 0$$

Note that the second derivative of this cubic is a line.

$$f''(x) = c\left(6x - 2(r_1 + r_2 + r_3)\right)$$

Setting this equal to zero and solving, we have

\begin{align} c\left(6a - 2(r_1 + r_2 + r_3)\right)&= 0\\ 6a - 2(r_1 + r_2 + r_3) &= 0 \tag{constant $c$ is non-zero}\\ 6a &= 2(r_1 + r_2 + r_3)\\ a &= \frac{r_1 + r_2 + r_3}{3} \end{align}

$$\square$$