Answer
$L(x)=\frac{x}{12}+\frac{4}{3}$
Work Step by Step
$f(x)=(x)^{\frac{1}{3}}$,
$L(x)=f(a)+f'(a)(x-a)$
$f'(x)=\frac{1}{3}{(x)^{\frac{-2}{3}}}$
The value of$x_{0}$ is chosen close to 8.5; we choose the value of 8 for simplicity
$x_{0}=8$
$f(8)=(8)^{\frac{1}{3}}=2$
$f'(8)=\frac{1}{3}{(8)^{\frac{-2}{3}}}=\frac{1}{12}$
then $L(x)=2+\frac{1}{12}(x-8)=\frac{x}{12}+\frac{4}{3}$
$L(x)=\frac{x}{12}+\frac{4}{3}$
the final answer :$L(x)=\frac{x}{12}+\frac{4}{3}$
