Find the linearization L (x) of the function at a.
$$f ( x ) = x ^ { 2 / 3 } ; \quad a = 64$$
A) $\frac { 8 } { 3 } x + \frac { 464 } { 3 }$
B) $\frac { 8 } { 3 } x + \frac { 512 } { 3 }$
C) $\frac { 8 } { 3 } x - \frac { 512 } { 3 }$
D) $\frac { 8 } { 3 } x - \frac { 464 } { 3 }$

Question

Quizplus · Accepted Answer

The linearization of a function at a point a is given by:
$$L(x)=f(a)+f'(a)(x-a)$$
Here, $f(x) = x^{2/3}$, so $f'(x) = \frac23x^{-1/3}$.
Substituting $a=64$ into the expression for $f'(x)$, we have $f'(64) = \frac23 \cdot 64^{-1/3} = \frac{2}{24} = \frac{1}{12}$.
Then, we have:
$$L(x) = f(64) + f'(64)(x-64) = 4 + \frac{1}{12}(x-64) = \frac{1}{12}x - \frac{464}{3}$$
Thus, the best choice is D.

Find the Linearization L (X) of the Function at A f