As $y=-\frac{1}{3} x^{3}+3 x$ is a cubic parabola, the line $e$ passing through its inflection point intersects the curve at two more points. The line, the ordinates of the intersection points, and the $X$-axis form two right-angled triangles. Let's examine the variation of the sum of the areas of these two triangles when they are located within the region bounded by the curve and the $X$-axis, and $e$ rotates in the counterclockwise direction.
First, let's examine the shape of the curve. The first derivative of the curve $y=-\frac{1}{3} x^{3}+3 x$ is $y^{\prime}=-x^{2}+3$, and the second derivative is $y^{\prime \prime}=-2 x$. The curve has an inflection point at $x=0, y=0$. The curve is symmetric with respect to the inflection point, i.e., the origin. If $x=-\infty, y=+\infty$; if $x=+\infty, y=-\infty$. The function decreases from $+\infty$ to a minimum at $x=-\sqrt{3}$, then increases to the inflection point at $x=0$, and continues to increase to a maximum at $x=+\sqrt{3}$. After that, it decreases to $-\infty$. The tangent at $x=0$ has a slope of $y_{x=0}^{\prime}=3$.

![](https://cdn.mathpix.com/cropped/2024_05_02_85fa1c46e5d1263bbdbeg-1.jpg?height=578&width=542&top_left_y=421&top_left_x=781)

The line passing through the inflection point, i.e., the origin, has the equation $y=m x$. This line intersects the curve at points other than the inflection point, which satisfy the equation:

$$
m x=-\frac{1}{3} x^{3}+3 x \quad \text { or } \quad x^{3}+(3 m-9) x=0
$$

This equation is satisfied by $x_{1}=0$ and the roots of the equation:

$$
x^{2}+(3 m-9)=0
$$

The roots are $x_{2,3}= \pm \sqrt{9-3 m}$. These two roots determine points on the curve that are symmetric with respect to the origin; for our purposes, it is sufficient to focus on the positive root. Indeed, the right triangles corresponding to these two points are congruent.

The ordinate of the point corresponding to the abscissa $x_{2}=\sqrt{9-3 m}$ is: $y_{2}=m \sqrt{9-3 m}$. The sum of the areas of the two right triangles is: $T=2 \frac{x_{2} y_{2}}{2}=m(9-3 m)$.

Clearly, $m$ can only take non-negative values, as long as $\sqrt{9-3 m}$ is real, i.e., $0<m<3$. The value $m=3$ is the slope of the tangent at the inflection point: this is the limit up to which the line (starting from the position $m=0$) can be rotated around the origin while still having a common point with the curve (other than the origin).

$T$ is a quadratic function of $m$ that increases from 0 to a maximum in the given interval, then decreases to zero. The maximum of $T$ occurs when $m=\frac{9}{6}=1.5$. At this point, $T_{\max }=6.75$. The abscissa corresponding to $m=1.5$ is $x=\sqrt{9-4.5}=$ $=1.5 \sqrt{2}$. (The point on the curve corresponding to this abscissa is $x=1.5 \sqrt{2}, y=2.25 \sqrt{2}$.)

Zoltán Czipott (Kegyesrendi g. VII. o. Szeged.)[^0]


[^0]:    i.e., the inflection point.