x | f'(x) | f"(x) | f'''(x) |
---|---|---|---|

0 | -1 | 2 | -6 |

1 | -1/2 | 1/4 | 3/8 |

Let $f(x)$ be a decreasing function with values for the first, second, and third derivatives evaluated at $x = 0$ and $x = 1$ given in the table shown above.

Given that the third derivative is strictly increasing on the interval $[0,1]$, if a second degree Taylor Polynomial centered at $x = 0$ is used to approximate $f(1)$, find the estimated error using Lagrange error bound.

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