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