1. The teacher said: "To place the largest possible circle inside a triangle with side lengths of $2$, $2$, and $2x$, what should the value of the positive real number $x$ be?"
Student A said: "I think $x=1$."
Student B said: "I believe $x=\sqrt{2}$."
Student C said: "Your answers are both incorrect."
Who among the three students is correct? Why?
1. On one hand, the area of the triangle is equal to $r(2+x)$; on the other hand, the height of the triangle on its base is $\sqrt{4-x^{2}}$, so the area of the triangle is equal to $x \sqrt{4-x^{2}}$. Therefore, $r=\frac{x \sqrt{4-x^{2}}}{2+x}$.
When $x=1$, $r=\frac{1}{\sqrt{3}}\frac{16}{27}>\frac{1}{1.7}$.
Thus, $x=\frac{4}{3}$ is a better choice.
Therefore, Student C's answer is correct.
Note: When $x=\sqrt{5}-1$, the maximum value of $r$ can be obtained as
$$
\frac{\sqrt{2}}{4}(\sqrt{5}-1)^{\frac{5}{2}} \text {. }
$$