5. let f be the function defined by f(x) = 4x³ - 5x + 3. which of the following is an equation of the line…

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=4x^{3}-5x + 3) is (f^{\prime}(x)=12x^{2}-5) (using the power rule ((x^{n})^\prime=nx^{n - 1})).

Step2: Find the slope of the tangent line at (x=-1)

Substitute (x = - 1) into (f^{\prime}(x)): (f^{\prime}(-1)=12(-1)^{2}-5=12 - 5=7).

Step3: Find the (y) - coordinate of the point on the function at (x=-1)

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

Step4: Use the point - slope form (y - y_{1}=m(x - x_{1}))

Here (m = 7), (x_{1}=-1), (y_{1}=4). The equation is (y - 4=7(x + 1)), which simplifies to (y=7x+11).

Answer:

(y = 7x + 11)