8 mark for review the graph of f, the derivative of a function f, is shown above. the points (2,6) and…

Explanation:

Step1: Recall tangent - line formula

The equation of the tangent line to the graph of (y = f(x)) at the point ((x_0,y_0)) is (y - y_0=f^{\prime}(x_0)(x - x_0)), where (f^{\prime}(x_0)) is the slope of the tangent line and ((x_0,y_0)) is the point of tangency.

Step2: Identify the point of tangency

We are given that the point of tangency is (x = 2), and the point ((2,6)) is on the graph of (f), so (x_0 = 2) and (y_0=6).

Step3: Find the slope of the tangent line

The slope of the tangent line to the graph of (y = f(x)) at (x = 2) is (f^{\prime}(2)). From the graph of (f^{\prime}), when (x = 2), we can estimate the value of (f^{\prime}(2)) (the (y) - value of the graph of (f^{\prime}) at (x = 2)). Since the graph of (f^{\prime}) is not clearly labeled with a value at (x = 2), we can also use the fact that the slope of the tangent line of (y = f(x)) at (x = 2) can be found using the two - point formula for the derivative (if we assume the function is differentiable and we can approximate). However, we can also use the fact that the equation of a line in slope - intercept form is (y=mx + b), and we know a point ((x_0,y_0)=(2,6)) on the line. We substitute (x = 2) and (y = 6) into each of the given equations of the lines (y=mx + b) (in the form (y - y_0=m(x - x_0)) or (y=mx+(b))) and check the slope (m). For option A: (y = 2x+1), when (x = 2), (y=2\times2 + 1=5\neq6), so A is incorrect. For option B: (y = 5x-4), when (x = 2), (y=5\times2-4 = 6). The slope (m = 5). For option C: (y = 5x - 10), when (x = 2), (y=5\times2-10=0\neq6), so C is incorrect. For option D: (y = 6x-6), when (x = 2), (y=6\times2-6 = 6), but we know that the slope of the tangent line (from the general concept of the derivative graph, if we assume a non - zero slope and the shape of the derivative graph) and by substituting the point ((2,6)) and checking the slope, the slope of the tangent line should be such that when we use the point - slope form (y - y_0=m(x - x_0)), the correct equation is (y-6 = 5(x - 2)) which simplifies to (y=5x-4).

Answer:

B. (y = 5x-4)