Question

If f(x) = 3x ^ 2 - 7x + 5 , find f^ prime (-4) .

Answer 1

Given:

$f(x)=3x^2-7x+5$

To determine f'(-4), we first find the derivative of the given equation as shown below:

$\begin{gathered} f(x)=3x^{2}-7x+5 \\ f^{\prime}(x)=6x-7 \end{gathered}$

Next,we plug in x=-4 into f'(x)=6x-7:

$\begin{gathered} f^{\prime}\left(x\right)=6x-7 \\ f^{\prime}(-4)=6(-4)-7 \\ Calculate \\ f^{\prime}(-4)=-31 \end{gathered}$

Hence, the value of f'(-4) is equal to -31.

Then, we also note that we can find the slope by plugging in x=-4 into f'(x)=6x-7.

Hence, the value of m is: -31

$m=-31$

Then,we use the point slope form:

$y-y_1=m(x-x_{1_})$

where:

m=slope=-31

(x1,y1)=point=(-4,81)

We plug in what we know:

$\begin{gathered} y-y_1=m(x-x_{1_)} \\ y-81=-31(x-(-4)) \\ Simplify\text{ and rearrange} \\ y-81=-31(x+4) \\ y-81=-31x-124 \\ y=-31x-124+81 \\ y=-31x-43 \end{gathered}$

Since the equation of the tangent line can be written in y=mx+b, the value of b is equal to -43.

Therefore, b is: -43