If the tangent line to y = f(x) at (7, 6) passes through the point (0, 5), find f(7) and f'(7)

To solve this problem, we start by determining the slope of the tangent line using the two points provided: (7, 6) and (0, 5).

The slope (m) of the line can be calculated using the formula:

m = (y2 – y1) / (x2 – x1) = (5 – 6) / (0 – 7) = -1 / -7 = 1/7

This slope implies that the derivative of the function f at x = 7, denoted as f'(7), is equal to 1/7.

Next, we have the point (7, 6) on the function, which directly gives us the value of the function f at that point. Therefore, we can conclude:

f(7) = 6

Now, we can summarize our findings:

• f(7) = 6
• f'(7) = 1/7