Calculus demonstration. Inflex points. Demonstrate that all cubic function with three different real zeros, has an inflex point whose coordinate x is the average of the three zeros. I've been meddling with this problem for a while, but I have not been able to even get close. I don't understand it, but I centainly understand the concepts that are mentioned.

Let's call the three different real roots of our cubic $a,b,c$. We can write our cubic as $(x-a)(x-b)(x-c)=(x^2-(a+b)x+ab)(x-c)=x^3-(a+b+c)x^2+(ab+ac+bc)x-abc$ Now the derivative of this cubic is $3x^2-2(a+b+c)x+ab+ac+bc$ and the second derivative is $6x-2(a+b+c)$. We have that in a point of inflection the second derivative is $=0$ thus this point has $x$ coordinate equal to $\frac{a+b+c}{3}$