Question

If the equation of f(x) goes through (1, 4) and (4, 6), what points does f′(x) go through?

A) (1, 4) and (4, 6)
B) (1, 6) and (4, 4)
C) (1, 0) and (4, 2)
D) (1, 2) and (4, 0)

Answer 1

The derivative f'(x) represents the slope of f(x). Calculating the slope between points (1, 4) and (4, 6) gives a constant slope of 2/3, which makes option D correct: (1, 2) and (4, 2).

Step-by-step explanation:

To determine the points through which the derivative of a function, f'(x), goes, one must first understand that f'(x) represents the slope of the function f(x) at each point. So, we are looking for the slope of the line that passes through the points (1, 4) and (4, 6). This means we calculate the change in y divided by the change in x. The change in y is 6 - 4 = 2, and the change in x is 4 - 1 = 3, so the slope is 2/3. Therefore, f'(x) will be constant (since the original function is linear) and equal to 2/3 at all points, meaning the points (1, 0) and (4, 2) are incorrect because they don't reflect a constant slope of 2/3. Given the options provided, the only possible correct answer would be (1, 2) and (4, 2), which means option D is correct. Each x-value is paired with the slope of f(x) at that x-value.