To solve this problem, we need to use the information given about the tangent line to the function f(x) at the point (5, 2) and the point (0, 1) through which the tangent line passes.
Firstly, we know that the point (5, 2) means that f(5) = 2. So, we have already found f(5) = 2.
Next, we can determine the slope of the tangent line, which is found by calculating the slope between the two points (5, 2) and (0, 1). The formula for the slope between two points (x1, y1) and (x2, y2) is:
m = (y2 – y1) / (x2 – x1)
In this context:
- (x1, y1) = (5, 2)
- (x2, y2) = (0, 1)
Substituting these values into the slope formula, we have:
m = (1 – 2) / (0 – 5) = -1 / -5 = 1/5
The slope of the tangent line at the point (5, 2) is also equal to the derivative of the function at that point, so we find that:
f'(5) = 1/5
In conclusion, we have:
- f(5) = 2
- f'(5) = 1/5