Let's plot the function $y=x^{x}$ in the interval $0<x \leq 1.5$. Determine the roots of the following equation by systematic trial, with at least two significant figures:

$$
x^{2 x}-2.05 x^{x}+1=0 .
$$
I. Having chosen $x$, we calculate $y$ using logarithms, for example, for $x=0.01$, $\lg y = x \lg x = 0.01 \cdot (-2) = -0.02 = 0.9800 - 1$, thus $y = 0.955$ (an approximate value with three decimal places). For the graph, we used the following value pairs:

| $x$ | $y$ | $x$ | $y$ | $x$ | $y$ |
| :--- | :---: | :---: | :---: | :---: | :---: |
| 0.01 | 0.955 | 0.35 | 0.693 | 0.9 | 0.909 |
| 0.03 | 0.900 | 0.4 | 0.693 | 1 | 1 |
| 0.05 | 0.861 | 0.5 | 0.707 | 1.1 | 1.110 |
| 0.1 | 0.794 | 0.6 | 0.736 | 1.2 | 1.245 |
| 0.2 | 0.725 | 0.7 | 0.779 | 1.3 | 1.406 |
| 0.3 | 0.697 | 0.8 | 0.837 | 1.4 | 1.602 |
|  |  |  |  | 1.5 | 1.837 |

We connected the points representing these value pairs with a smooth curve that fits them and accepted this as the graph of our function. The graph sinks from $x=0.01$ to approximately $x=0.35$, initially faster, then more slowly, and then rises from about 0.4 to 1.5, initially slowly, then faster.

![](https://cdn.mathpix.com/cropped/2024_05_02_9477db57bd77cf511043g-1.jpg?height=824&width=1308&top_left_y=772&top_left_x=404)

II. The given equation is a quadratic equation in terms of $y = x^x$:

$$
y^2 - 2.05y + 1 = 0
$$

from which $y_1 = 0.8$ and $y_2 = 1.25$.

To find an approximate solution to the equation $x^x = 0.8$, we can use our graph. The line $y_1 = 0.8$ intersects the curve at two points: the abscissa of the first point is a number between $x=0.05$ and $x=0.1$, much closer to the latter, and the abscissa of the second point is between $x=0.7$ and $x=0.8$, somewhat closer to the former.

To find a more accurate approximate value, we use linear interpolation. The curve's decline between the points with abscissae 0.05 and 0.1 is 0.067, and the decline from 0.861 to 0.8 is 0.061, which is about $9/10$ of the former. Therefore, we expect the root to be close to $x_1 = 0.095$. Here, the value of $y$ (from $\lg y = 0.9029 - 1$) is $0.7997 \approx 0.8$. Further refinement using a four-digit function table is not expected.

Similarly, between the points with abscissae 0.7 and 0.8, the rise is 0.058, and we are looking for a rise of 0.021 from the former point, which is nearly 4/10 of the total rise (we round up because the curve initially rises more slowly), so the abscissa of the intersection point is approximately $x_2 = 0.74$. This is consistent because $0.74^{0.74} \approx 0.8002$.

The line $y_2 = 1.25$ intersects the graph at one point, and the approximate value of the root is similarly $x_3 = 1.204$.

Thus, our equation has one root between 0.09 and 0.10, another around 0.74, and a third between 1.20 and 1.21. (It can be shown that there are no further roots for $x > 1.5$.)