find an equation of the tangent line to the graph of ( y = g(x) ) at ( x = 6 ) if ( g(6)=-4 ) and (…

Explanation:

Step1: Recall the point - slope form of a line

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)) is a point on the line and (m) is the slope of the line.

Step2: Identify the values of (x_1,y_1) and (m)

We are given that (x = 6), (g(6)=-4), so (x_1 = 6) and (y_1=-4). Also, (g^{\prime}(6) = 2), and the slope (m) of the tangent line at (x = 6) is (m = g^{\prime}(6)=2).

Step3: Substitute the values into the point - slope form

Substitute (x_1 = 6), (y_1=-4) and (m = 2) into (y - y_1=m(x - x_1)). We get (y-(-4)=2(x - 6)).

Step4: Simplify the equation

(y + 4=2x-12). Then (y=2x-16).

Answer:

(y = 2x-16)