Answer
Left hand $x_2\approx-0.42$
Right hand $x_2\approx2.42$
Work Step by Step
Step 1. Given $f(x)=-x^2+2x+1$, we have $f'(x)=-2x+2$. Starting with $x_0=0$ for the left hand, we can use Newton’s method $x_{n+1}=x_n-\frac{f(x)}{f'(x)}$ to estimate the solution up to $x_2$ as shown in the table. The result gives $x_2\approx-0.42$ for this case.
Step 2. Repeat the above step for the right hand with $x_0=2$; we can get the result $x_2\approx2.42$ for this case as shown in the table.
