Question

find the linearization ( l(x) ) at ( x = a ).
( g(x)=2 x^{2}-5 x-4 quad a=-3 )
( l(x)= )

Explanation:

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

Given \(f(x)=2x^{2}-5x - 4\) and \(a=-3\).
Substitute \(x = a=-3\) into \(f(x)\):
\(f(-3)=2\times(-3)^{2}-5\times(-3)-4\)
\(=2\times9 + 15-4\)
\(=18 + 15-4=29\)

Step2: Find the derivative of \(f(x)\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), for \(f(x)=2x^{2}-5x - 4\), \(f^\prime(x)=(2x^{2})^\prime-(5x)^\prime-(4)^\prime\)
\(f^\prime(x)=4x-5\)

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

Substitute \(x = a=-3\) into \(f^\prime(x)\):
\(f^\prime(-3)=4\times(-3)-5=-12 - 5=-17\)

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

Here \(a=-3\), \(f(a) = 29\), \(f^\prime(a)=-17\)
\(L(x)=29-17(x+3)\)
Expand the expression:
\(L(x)=29-17x-51=-17x - 22\)

Answer:

\(L(x)=-17x - 22\)