choose from the options below the linearization of: f(x)=x³ at x = 2 y = 8x+12 y = 12x - 16 y = 8x - 4 y =…

Explanation:

Step1: Recall linearization formula

The linearization $L(x)$ of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$.

Step2: Find $f(2)$

Given $f(x)=x^{3}$, then $f(2)=2^{3}=8$.

Step3: Find the derivative of $f(x)$

Using the power - rule $(x^{n})^\prime=nx^{n - 1}$, for $f(x)=x^{3}$, $f^{\prime}(x)=3x^{2}$.

Step4: Find $f^{\prime}(2)$

Substitute $x = 2$ into $f^{\prime}(x)$, so $f^{\prime}(2)=3\times2^{2}=12$.

Step5: Calculate the linearization

Substitute $a = 2$, $f(2)=8$ and $f^{\prime}(2)=12$ into the linearization formula $L(x)=f(a)+f^{\prime}(a)(x - a)$. We get $L(x)=8 + 12(x - 2)=8+12x-24=12x - 16$.

Answer:

$y = 12x - 16$