Final answer:
The slope of the tangent line, represented as f'(6), is 1/6 and the value of the function f at x=6, denoted f(6), is 3.
Step-by-step explanation:
If the tangent line to y = f(x) at the point (6, 3) passes through the point (0, 2), we can find f(6) and f'(6). Since we know a point on the curve and a point the tangent passes through, we can determine the slope of the tangent line, which is also f'(6), the derivative of the function at x=6.
First, we calculate the slope using the two given points:
Slope, m = (y2 - y1) / (x2 - x1) = (2 - 3) / (0 - 6) = -1 / -6 = 1/6.
So, f'(6) = 1/6. The value of f(6) is simply the y-coordinate of the point on the curve,
which is given as 3. Therefore, f(6) = 3.