Did You Know what Actually is Point of Inflection ? (Part-1)

Visually, the lines aren't tangents at all... and while the problem is nice and a graph helps a lot, it doesn't count as a proof (I'd say that the intersection point is at 7.something, not 8, from the graph).

This is my proof with lengthy easy sections omitted:

The tangent to $f(x)$ in $x_1$ is a line described by $y=f(x_1)+f'(x_1)(x-x_1)$ , since it has to coincide with $f(x)$ in $x_1$ and have the same first derivative (slope). We're looking for the highest point above $f(x)$ in which two tangents intersect.

Let us assume we already proved (rigorously, not "proved" by picture) that $x_0=0$ and are already looking just for the highest such point. Then, for any $x_1$ , the tangent intersects the $y$ -axis in $y_1(x_1)=f(x_1)-x_1f'(x_1)$ . Again, we take a first derivative: $\frac{\mathrm{d}y_1}{\mathrm{d}x_1}=f'(x_1)-(f'(x_1)+x_1f''(x_1))=-x_1f''(x_1)$ . In the maximum, $f''(x_1)=0$ or $x_1=0$ . The latter choice is obviously wrong, since there's a local maximum of $f(x)$ in $x=0$ . That leaves us with $f''(x_1)=4x_1(x_1^2-1)$ , or $x_1=\pm1$ . These are two inflection points, whose tangents intersect at $y_0=f(1)-f'(1)=8$ .