The slope of a function f(x) is known to be 0 at point x*. The function may not attain extremum point at x*.  Select one:TrueFalse

Analysis

1. Understanding the Slope: The slope of a function at a point indicates the rate of change of the function at that point. A slope of 0 suggests that the function is neither increasing nor decreasing at that point.

2. Extremum Point Condition: An extremum (maximum or minimum) occurs at a point where the slope goes from positive to negative (maximum) or from negative to positive (minimum). However, a point where the slope is 0 does not necessarily imply that the function has an extremum.

3. Possible Scenarios:

   • The function could have a horizontal tangent line (indicating a possible extremum).
   • The function could also have a point of inflection, where the slope is 0 but the function does not attain a maximum or minimum (e.g., the function could change its concavity).

Final Answer

True: The function may not attain an extremum point at x*, despite having a slope of 0.