Questions: The table below gives values of a differentiable function (f(x)). x012345678910 ------------ f(x)-5-21-1-3-212467 Estimate the (x)-values of critical points of (f(x)) on the interval (0<x<10) and classify them below. (f(x)) has relative maximum(s) at (x=) (f(x)) has relative minimum(s) at (x=) (f(x)) has critical point(s) that are not extrema at (x=)

Solution Steps

To estimate the critical points of the function \( f(x) \), we need to identify where the derivative \( f'(x) \) changes sign. This typically occurs at points where \( f(x) \) changes from increasing to decreasing (indicating a relative maximum) or from decreasing to increasing (indicating a relative minimum). We can approximate the derivative by examining the differences between consecutive \( f(x) \) values. A critical point that is not an extremum occurs when the derivative does not change sign.

We calculated the differences between consecutive values of \( f(x) \) to approximate the derivative \( f'(x) \). The differences are given by: \[ \text{differences} = [3, 3, -2, -2, 1, 3, 1, 2, 2, 1] \]

A relative maximum occurs where the derivative changes from positive to negative. From the differences, we observe:

• At \( x = 2 \), the difference changes from positive (3) to negative (-2). Thus, we have a relative maximum at: \[ f(x) \text{ has a relative maximum at } x = 2 \]

A relative minimum occurs where the derivative changes from negative to positive. From the differences, we find:

• At \( x = 4 \), the difference changes from negative (-2) to positive (1). Therefore, we have a relative minimum at: \[ f(x) \text{ has a relative minimum at } x = 4 \]

We check for critical points that are not extrema, which occur when the derivative does not change sign. In this case, there are no such points identified: \[ f(x) \text{ has critical points that are not extrema at } x = [] \]

Final Answer