Imagine the segment of the $X$-axis between points $x_{1}$ and $x_{2}$ as a weightless rod cut out from the coordinate plane; place a weight $p_{1}$ at the $x_{1}$ endpoint, and a weight $p_{2}$ at the $x_{2}$ endpoint. At which point on the $X$ axis will the center of mass of the rod be located?

Place a weight $p_{1}$ at the point $\left(x_{1}, y_{1}\right)$ on the curve $y=f(x)$, and a weight $p_{2}$ at the point $\left(x_{2}, y_{2}\right)$. If we imagine these points connected by a weightless rod, where will the center of mass of the rod be located now? (What will be its abscissa and what will be its ordinate?)

What does the Jensen inequality mean "physically"? What happens if the weights $p_{1}$ and $p_{2}$ are equal?
We used the terms "weight" and "weighted," and indeed the expressions for the roles resemble those in the expression for the center of mass. Place a weight $p_{1}$ at the point $x_{1}$ on the $X$-axis, and a weight $p_{2}$ at the point $x_{2}$. Imagine these are connected by a weightless rod. The center of mass of the rod divides it in the inverse ratio of the placed weights, that is, at the point $x$ where $\frac{x-x_{1}}{x_{2}-x}=\frac{p_{2}}{p_{1}}$. From this,

$$
\begin{gathered}
x=\frac{p_{1} x_{1}+p_{2} x_{2}}{p_{1}+p_{2}}=q_{1} x_{1}+q_{2} x_{2}, \quad \text { where } \quad q_{1}=\frac{p_{1}}{p_{1}+p_{2}} \\
q_{2}=\frac{p_{2}}{p_{1}+p_{2}}, \quad q_{1}+q_{2}=1
\end{gathered}
$$

If, however, we place a weight $p_{1}$ at the point $\left(x_{1}, y_{1}\right)=\left(x_{1}, f\left(x_{1}\right)\right)$ on a curve $f(x)$, and a weight $p_{2}$ at the point $\left(x_{2}, y_{2}\right)=\left(x_{2}, f\left(x_{1}\right)\right)$, then performing a similar calculation for the ordinate (and using the above notation), it follows that the coordinates of the center of mass are

$$
\left(q_{1} x_{1}+q_{2} x_{2} ; q_{1} y_{1}+q_{2} y_{2}\right)=\left(q_{1} x_{1}+q_{2} x_{2} ; q_{1} f\left(x_{1}\right)+q_{2} f\left(x_{2}\right)\right)
$$

The Jensen inequality thus expresses that if weights are placed on two points of a convex curve, then the center of mass of the weights is above the curve.