Now assume that the table gives values of the continuous function y=f′(x) (instead of f(x)). Estimate and classify critical points of the function f(x).

Question

The following table gives values of the differentiable function .

x	0	1	2	3	4	5	6	7	8	9	10
y	-1	-3	-5	-2	1	2	-1	-2	1	2	5

Now assume that the table gives values of the continuous function  (instead of ). Estimate and classify critical points of the function.

critical points and classifications:

Answer 1

Critical points occur where f'(x) = y = 0. Therefore, we will have a critical point for x such that 3<x<4, 5<x<6, and 7<x<8. When x is between 3 and 4, the critical point is a local minimum because the derivative changes from negative to positive. The same holds true for when x is between 7 and 8. When x is between 5 and 6, the critical point is a local maximum because the derivative changes from positive to negative.