$ma{x}_{0}x1\mathrm{.}2|f^{(5)}(x)|=8\mathrm{.}4,ma{x}_{0}x1\mathrm{.}2|f^{(6)}(x)|=58\mathrm{.}8,ma{x}_{0}x1\mathrm{.}2|f^{(7)}(x)|=411\mathrm{.}8$

Let $P(x)$ be the fifth$-$degree Taylor polynomial for a function $f$ about $x=0.$ Information about the maximum of the absolute value of selected derivatives of $f$ over the interval $0x1\mathrm{.}2$ is given in the table above. Of the following, which is the smallest value of $k$ for which the Lagrange error bound guarantees that $|f(1\mathrm{.}2)-P(1\mathrm{.}2)|k?$

$0\mathrm{.}174$

$0\mathrm{.}244$

$0\mathrm{.}293$