Question

find the linearization ( l(x) ) at ( x = a ).

( f(x)=-2 x^{3}+5 x + 1 quad a = 2 )

( l(x)=square )

Explanation:

Step1: Find the value of \( f(a) \)

Substitute \( x = a = 2 \) into \( f(x)=-2x^{3}+5x + 1 \).

$$
LATEXBLOCK0
$$

Step2: Find the derivative \( f^{\prime}(x) \)

Differentiate \( f(x)=-2x^{3}+5x + 1 \) using the power rule \( (x^{n})^\prime=nx^{n - 1} \).
\( f^{\prime}(x)=-2\times3x^{2}+5=-6x^{2}+5 \)

Step3: Find the value of \( f^{\prime}(a) \)

Substitute \( x = a = 2 \) into \( f^{\prime}(x) \).

$$
LATEXBLOCK1
$$

Step4: Use the linearization formula \( L(x)=f(a)+f^{\prime}(a)(x - a) \)

Substitute \( f(2)=-5 \), \( f^{\prime}(2)=-19 \), and \( a = 2 \) into the formula.

$$
LATEXBLOCK2
$$

Answer:

\( L(x)=-19x + 33 \)