Question

find the linearization ( l(x) ) at ( x = a ).
( f(x)=-4 x^{3}-5 x + 1 quad a = 3 )
( l(x)= )

Explanation:

Step1: Find \( f(a) \)

Substitute \( x = a = 3 \) into \( f(x)=-4x^{3}-5x + 1 \).

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Step2: Find \( f^{\prime}(x) \) and \( f^{\prime}(a) \)

Differentiate \( f(x)=-4x^{3}-5x + 1 \) using the power rule \( (x^{n})^\prime=nx^{n - 1} \).
\( f^{\prime}(x)=-12x^{2}-5 \)
Substitute \( x = 3 \) into \( f^{\prime}(x) \):
\( f^{\prime}(3)=-12\times(3)^{2}-5=-12\times9-5=-108 - 5=-113 \)

Step3: Use the linearization formula \( L(x)=f(a)+f^{\prime}(a)(x - a) \)

Here \( a = 3 \), \( f(a)=-122 \), \( f^{\prime}(a)=-113 \)

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Answer:

\( L(x)=-113x + 217 \)