3. Let $f(x) = e^{-2x} - e^x + 1$. Suppose one wants to find a root of the equation $f(x) = 0$.
(a) (7 points) Show that the interval $[0, 1]$ contains a root.
(b) (7 points) Apply one step of the Newton-Raphson algorithm with starting point $x_0 = 1$ to give $x_1$. Also apply one step of the Newton-Raphson algorithm with starting point $y_0 = 0$ to give $y_1$.
(c) (7 points) Verify whether the interval $(y_1, x_1)$ is contained in $(0, 1)$ and whether it contains the root.