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 f(x) at x=a is given by:

$$L(x) = f(a) + f'(a)(x-a)$$

First, we need to find f(a) and f'(a):

$$f(64) = 64^{2/3} = 16$$

$$f'(x) = \frac{2}{3}x^{-1/3}$$

$$f'(64) = \frac{2}{3}(64)^{-1/3} = \frac{2}{3}\cdot\frac{1}{4} = \frac{1}{6}$$

Now we can plug these values into the linearization formula:

$$L(x) = 16 + \frac{1}{6}(x-64) = \frac{1}{6}x + \frac{464}{3}$$

So the correct answer is D, with the linearization given by:

$$L(x) = \frac{8}{3}x - \frac{464}{3}$$

Find the Linearization L(x)L ( x )L(x) Of the Function At aaa

Find the Linearization $L ( x )$ Of the Function At $a$